mathai 1.1.1

Mathematics solving Ai tailored to NCERT

Math AI Documentation

Source

Github repository of the code https://github.com/infinity390/mathai4

Philosophy

I think it is a big realization in computer science and programming to realize that computers can solve mathematics.
This understanding should be made mainstream. It can help transform education, mathematical research, and computation of mathematical equations for work.

Societal Implications Of Such A Computer Program And The Author's Comment On Universities Of India

I think mathematics is valued by society because of education. Schools and universities teach them.
So this kind of software, if made mainstream, could bring real change.

The Summary Of How Computer "Solves" Math

Math equations are a tree data structure (TreeNode class).
We can manipulate the math equations using various algorithms (functions provided by the mathai library).
We first parse the math equation strings to get the tree data structure (parse function in mathai).

The Library

Import the library by doing:

from mathai import *

str_form

It is the string representation of a TreeNode math equation.

Example

(cos(x)^2)+(sin(x)^2)

Is represented internally as:

f_add
 f_pow
  f_cos
   v_0
  d_2
 f_pow
  f_sin
   v_0
  d_2

Leaf Nodes

Variables (start with a v_ prefix):

• v_0 -> x
• v_1 -> y
• v_2 -> z
• v_3 -> a

Numbers (start with d_ prefix; only integers):

• d_-1 -> -1
• d_0 -> 0
• d_1 -> 1
• d_2 -> 2

Branch Nodes

• f_add -> addition
• f_mul -> multiplication
• f_pow -> power

parse

Takes a math equation string and outputs a TreeNode object.

from mathai import *

equation = parse("sin(x)^2+cos(x)^2")
print(equation)

Output

(cos(x)^2)+(sin(x)^2)

simplify

It simplifies and cleans up a given math equation.

from mathai import *

equation = simplify(parse("(x+x+x+x-1-1-1-1)*(4*x-4)*sin(sin(x+x+x)*sin(3*x))"))
printeq(equation)

Output

((-4+(4*x))^2)*sin((sin((3*x))^2))

Incomplete Documentation, Will be updated and completed later on

Demonstrations

Example Demonstration 1 (derivation of hydrogen atom's ground state energy in electron volts using the variational principle in quantum physics)

from mathai import *
basic_int = lambda x: dowhile(x, lambda y: fraction(simplify(integrate_const(integrate_formula(integrate_summation(expand(y)))))))
algebra = lambda x: dowhile(x, lambda y: fraction(simplify(y)))
z =  simplify(parse("1"))
k =  simplify(parse("8987551787"))
m =  simplify(parse("9109383701 * 10^(-40)"))
e1=  simplify(parse("1602176634 * 10^(-28)"))
hbar=simplify(parse("1054571817 * 10^(-43)"))
pi = tree_form("s_pi")
euler = tree_form("s_e")
r = parse("r")
a0 = hbar**2 / (k*e1**2*m)
c2 = z/a0
c1 = (z**3 / (pi * a0**3)).fx("sqrt")
psi = c1 * euler**(-c2 * r)
psi2 = psi**2
laplace_psi = diff(r**2 * diff(psi, r.name), r.name)/r**2
psi2 = simplify(psi2)
integral_psi2 = TreeNode("f_integrate", [psi2 * parse("4")* pi * r**2, r])
integral_psi2 = simplify(integral_psi2)
integral_psi2 = integrate_subs(integral_psi2)
integral_psi2 = basic_int(integral_psi2)
integral_psi2 = integrate_byparts(integral_psi2)
integral_psi2 = basic_int(integral_psi2)
integral_psi2 = integrate_byparts(integral_psi2)
integral_psi2 = basic_int(integral_psi2)
integral_psi2 = integrate_clean(integral_psi2)
integral_psi2 = algebra(integral_psi2)
a = limit1(TreeNode("f_limit", [integral_psi2, r]))
b = limit3(TreeNode("f_limitpinf", [integral_psi2, r]))
integral_psi2 = simplify(b-a)
V = -(k * z * e1**2)/r
Hpsi = -hbar**2/(2*m) * laplace_psi + V*psi
psiHpsi = psi * Hpsi
integral_psiHpsi = TreeNode("f_integrate", [psiHpsi * parse("4")* pi * r**2, r])
integral_psiHpsi = simplify(integral_psiHpsi)
integral_psiHpsi = integrate_subs(integral_psiHpsi)
integral_psiHpsi = basic_int(integral_psiHpsi)
integral_psiHpsi = integrate_byparts(integral_psiHpsi)
integral_psiHpsi = basic_int(integral_psiHpsi)
integral_psiHpsi = integrate_byparts(integral_psiHpsi)
integral_psiHpsi = basic_int(integral_psiHpsi)
integral_psiHpsi = integrate_clean(integral_psiHpsi)
integral_psiHpsi = algebra(integral_psiHpsi)
a = limit1(TreeNode("f_limit", [integral_psiHpsi, r]))
b = limit3(limit2(expand(TreeNode("f_limitpinf", [integral_psiHpsi, r]))))
integral_psiHpsi = simplify(b-a)
result =  integral_psiHpsi / integral_psi2
print(compute(result /e1))

Output

-13.605693122882867

Example Demonstration 2 (boolean algebra)

from mathai import *
eq = parse("(A-B)|(B-A)")
eq = set_sub(eq)
eq = solve_logically(eq)
print(eq)
eq = parse("(A|B)-(A&B)")
eq = set_sub(eq)
eq = solve_logically(eq)
print(eq)

Output

(~A&B)|(~B&A)
(~A&B)|(~B&A)

Example Demonstration 3 (limits approaching to a constant value)

from mathai import *
limits = ["(e^(tan(x)) - 1 - tan(x)) / x^2", "sin(x)/x", "(1-cos(x))/x^2", "(sin(x)-x)/sin(x)^3"]
for q in limits:
    q = fraction(simplify(TreeNode("f_limit",[parse(q),parse("x")])))
    q = limit1(q)
    print(q)

Output

1/2
1
1/2
-(1/6)

Example Demonstration 4 (limits approaching to infinity)

from mathai import *
eq= parse("limitpinf((3*x^2+x)/(2*x^2+5),x)")
eq = simplify(eq)
eq = limit4(eq)
eq = simplify(eq)
eq = limit3(eq)
eq = simplify(eq)
print(eq)

Output

3/2

Example Demonstration 5 (linear equations) (general solution of linear equations in two variables)

from mathai import *
eq= parse("a*x+b*y+c = 0 & d*x+f*y+g = 0")
eq = simplify(eq)
eq = linear_solve(eq, [parse(item) for item in "a b c d f g".split(" ")])
eq = factor1(eq)
eq = simplify(eq)
for item in eq.children:
  print(item)

Output

((((c*f)-(b*g))/((a*f)-(b*d)))+x)=0
((((a*g)-(c*d))/((a*f)-(b*d)))+y)=0

Example Demonstration 6 (expectation algebra)

from mathai import *
eq = simplify(parse("covariance(A+B,C+D)=covariance(B,C)+covariance(B,D)+covariance(A,D)+covariance(A,C)"))
eq = logic0(simplify(expect(simplify(expand(expect(eq))))))
print(eq)

Output

true

Questions solved using god() function

Code

from mathai import *
lst = [
"(A-B)|(B-A) <-> (A|B)-(A&B)", "(cosec(x)-cot(x))^2=(1-cos(x))/(1+cos(x))",
"cos(x)/(1+sin(x))+(1+sin(x))/cos(x)=2*sec(x)", "tan(x)/(1-cot(x))+cot(x)/(1-tan(x))=1+sec(x)*cosec(x)",
"(1+sec(x))/sec(x)=sin(x)^2/(1-cos(x))", "(cos(x)-sin(x)+1)/(cos(x)+sin(x)-1)=cosec(x)+cot(x)",
"(sin(x)-2*sin(x)^3)/(2*cos(x)^3-cos(x))=tan(x)", "(sin(x)+cosec(x))^2+(cos(x)+sec(x))^2=7+tan(x)^2+cot(x)^2",
"(cosec(x)-sin(x))*(sec(x)-cos(x))=1/(tan(x)+cot(x))", "abs(abs(x-2)-3)<=2",
"abs(x)>=0", "2*x/(2*x^2+5*x+2)>1/(x+1)",
"(x+2)*(x+3)/((x-2)*(x-3))<=1", "(5*x-1)<(x+1)^2&(x+1)^2<7*x-3",
"abs(x+5)*x+2*abs(x+7)-2=0", "x*abs(x)-5*abs(x+2)+6=0",
"x^2-abs(x+2)+x>0", "abs(3*x-5)+abs(8-x)=abs(3+2*x)",
"abs(x^2+5*x+9)<abs(x^2+2*x+2)+abs(3*x+7)",

"dif(y,x)=sqrt(4-y^2)", "dif(y,x)+y=1", "x^5*dif(y,x)=-y^5",
"dif(y,x)=(1+x^2)*(1+y^2)", "x*(x^2-1)*dif(y,x)=1", "dif(y,x)=arcsin(x)",
"(x^2+x*y)*dif(y,x)=(x^2+y^2)", "dif(y,x)=(x+y)/x",
"(x-y)*dif(y,x)-(x+y)=0", "(x^2-y^2)+2*x*y*dif(y,x)=0",
"x*dif(y,x)-y+x*sin(y/x)=0", "(x+y)*dif(y,x)=1",
"x^2*dif(y,x)=x^2-2*y^2+x*y",

"integrate(1/(1+sin(x)),x)", "integrate((7^(7^(7^x)))*(7^(7^x))*(7^x),x)",
"integrate(abs(x+1),x,-4,10)", "integrate(abs(sin(x)),x,-pi,pi/6)",
"integrate(2*x/(1+x^2),x)", "integrate(sqrt(a*x+b),x)", "integrate(x*sqrt(x),x)",
"integrate(x*sqrt(1+2*x^2),x)", "integrate(x/sqrt(x+4),x)",
"integrate(e^(2*x+3),x)", "integrate(x/e^(x^2),x)",
"integrate(sqrt(sin(2*x))*cos(2*x),x)", "integrate(sin(x)*sin(cos(x)),x)",
"integrate(sin(2*x+5)^2,x)", "integrate(sin(3*x)*cos(4*x),x)",
"integrate(cos(2*x)*cos(4*x)*cos(6*x),x)", "integrate(sin(2*x+1)^3,x)",
"integrate(sin(x)^3*cos(x)^3,x)", "integrate(sin(x)*sin(2*x)*sin(3*x),x)",
"integrate(sin(4*x)*sin(8*x),x)", "integrate(sin(x)^4,x)",
"integrate(cos(2*x)^4,x)", "integrate(x/((x+1)*(x+2)),x)",
"integrate(1/(x^2-9),x)", "integrate((3*x-1)/((x-1)*(x-2)*(x-3)),x)",
"integrate(x/((x-1)*(x-2)*(x-3)),x)", "integrate(2*x/(x^2+3*x+2),x)",
"integrate((1-x^2)/(x*(1-2*x)),x)", "integrate(x/((x-1)*(x^2+1)),x)",
"integrate(x/((x-1)^2*(x+2)),x)",
"integrate((2*x-3)/((x^2-1)*(2*x+3)),x)",
"integrate(5*x/((x+1)*(x^2-4)),x)",
"integrate(2/((1-x)*(1+x^2)),x)"
]
for s in lst:
    god(s)

Output

? (A-B)|(B-A) <-> (A|B)-(A&B)
thinking...
=> true

? (cosec(x)-cot(x))^2=(1-cos(x))/(1+cos(x))
thinking...
(-((1-cos(x))/(1+cos(x)))+((cosec(x)-cot(x))^2))=0
(-((1-cos(x))/(1+cos(x)))+(((1/sin(x))-(cos(x)/sin(x)))^2))=0
((cos(x)*(sin(x)^8))+((cos(x)^3)*(sin(x)^6))-(cos(x)*(sin(x)^6))-((cos(x)^2)*(sin(x)^6))-(sin(x)^8)+(sin(x)^6))=0
0=0
true
=> true

? cos(x)/(1+sin(x))+(1+sin(x))/cos(x)=2*sec(x)
thinking...
(((1+sin(x))/cos(x))+(cos(x)/(1+sin(x)))-(2*sec(x)))=0
(((1+sin(x))/cos(x))+(cos(x)/(1+sin(x)))-(2/cos(x)))=0
((cos(x)*(sin(x)^2))-cos(x)+(cos(x)^3))=0
0=0
true
=> true

? tan(x)/(1-cot(x))+cot(x)/(1-tan(x))=1+sec(x)*cosec(x)
thinking...
(-1+(cot(x)/(1-tan(x)))+(tan(x)/(1-cot(x)))-(cosec(x)*sec(x)))=0
(-1+((cos(x)/sin(x))/(1-(sin(x)/cos(x))))+((sin(x)/cos(x))/(1-(cos(x)/sin(x))))-(1/(cos(x)*sin(x))))=0
((2*cos(x)*(sin(x)^3))+(2*(cos(x)^3)*sin(x))-(2*cos(x)*sin(x))-(2*(cos(x)^2)*(sin(x)^2))-(cos(x)^4)-(sin(x)^4)+(cos(x)^2)+(sin(x)^2))=0
0=0
true
=> true

? (1+sec(x))/sec(x)=sin(x)^2/(1-cos(x))
thinking...
(((1+sec(x))/sec(x))-((sin(x)^2)/(1-cos(x))))=0
(((1+(1/cos(x)))/(1/cos(x)))-((sin(x)^2)/(1-cos(x))))=0
(1-(cos(x)^2)-(sin(x)^2))=0
0=0
true
=> true

? (cos(x)-sin(x)+1)/(cos(x)+sin(x)-1)=cosec(x)+cot(x)
thinking...
(((1+cos(x)-sin(x))/(-1+cos(x)+sin(x)))-cosec(x)-cot(x))=0
(((1+cos(x)-sin(x))/(-1+cos(x)+sin(x)))-(1/sin(x))-(cos(x)/sin(x)))=0
(-((cos(x)^2)*sin(x))-(sin(x)^3)+sin(x))=0
0=0
true
=> true

? (sin(x)-2*sin(x)^3)/(2*cos(x)^3-cos(x))=tan(x)
thinking...
(((-(2*(sin(x)^3))+sin(x))/((2*(cos(x)^3))-cos(x)))-tan(x))=0
(((-(2*(sin(x)^3))+sin(x))/((2*(cos(x)^3))-cos(x)))-(sin(x)/cos(x)))=0
((2*cos(x)*sin(x))-(2*cos(x)*(sin(x)^3))-(2*(cos(x)^3)*sin(x)))=0
0=0
true
=> true

? (sin(x)+cosec(x))^2+(cos(x)+sec(x))^2=7+tan(x)^2+cot(x)^2
thinking...
(-7-(cot(x)^2)-(tan(x)^2)+((cos(x)+sec(x))^2)+((cosec(x)+sin(x))^2))=0
(-7-((cos(x)/sin(x))^2)-((sin(x)/cos(x))^2)+((cos(x)+(1/cos(x)))^2)+(((1/sin(x))+sin(x))^2))=0
(((cos(x)^2)*(sin(x)^4))+((cos(x)^4)*(sin(x)^2))+((cos(x)^4)*(sin(x)^6))+((cos(x)^6)*(sin(x)^4))-(3*(cos(x)^4)*(sin(x)^4))-((cos(x)^2)*(sin(x)^6))-((cos(x)^6)*(sin(x)^2)))=0
0=0
true
=> true

? (cosec(x)-sin(x))*(sec(x)-cos(x))=1/(tan(x)+cot(x))
thinking...
(((cosec(x)-sin(x))*(-cos(x)+sec(x)))-(1/(cot(x)+tan(x))))=0
((((1/cos(x))-cos(x))*((1/sin(x))-sin(x)))-(1/((cos(x)/sin(x))+(sin(x)/cos(x)))))=0
((cos(x)*(sin(x)^3))+((cos(x)^3)*(sin(x)^5))+((cos(x)^3)*sin(x))+((cos(x)^5)*(sin(x)^3))-(3*(cos(x)^3)*(sin(x)^3))-(cos(x)*(sin(x)^5))-((cos(x)^5)*sin(x)))=0
0=0
true
=> true

? abs(abs(x-2)-3)<=2
thinking...
=> false

? abs(x)>=0
thinking...
=> false

? 2*x/(2*x^2+5*x+2)>1/(x+1)
thinking...
(((2*x)/(2+(2*(x^2))+(5*x)))-(1/(1+x)))>0
~((((2*x)/(2+(2*(x^2))+(5*x)))-(1/(1+x)))=0)&~((((2*x)/(2+(2*(x^2))+(5*x)))-(1/(1+x)))<0)
~((-2-(3*x))=0)&~(((-2-(3*x))/((1+x)*(2+(2*(x^2))+(5*x))))<0)
~((-2-(3*x))=0)&~(((-2-(3*x))/((1+x)*(2+x)*((1/2)+x)))<0)
(-2,-1)U(-(2/3),-(1/2))U{-1,-(1/2),-2}
=> (-2,-1)U(-(2/3),-(1/2))U{-1,-(1/2),-2}

? (x+2)*(x+3)/((x-2)*(x-3))<=1
thinking...
(-1+(((2+x)*(3+x))/((-2+x)*(-3+x))))<=0
((-1+(((2+x)*(3+x))/((-2+x)*(-3+x))))=0)|((-1+(((2+x)*(3+x))/((-2+x)*(-3+x))))<0)
(x=0)|((x/((-2+x)*(-3+x)))<0)
(-inf,0)U(2,3)U{0}
=> (-inf,0)U(2,3)U{0}

? (5*x-1)<(x+1)^2&(x+1)^2<7*x-3
thinking...
((-1+(5*x)+((-1-x)*(1+x)))<0)&((3-(7*x)+((1+x)^2))<0)
((-2+(3*x)-(x^2))<0)&((4-(5*x)+(x^2))<0)
(((-1+x)*(-4+x))<0)&(((-2+x)*(1-x))<0)
(2,4)
=> (2,4)

? abs(x+5)*x+2*abs(x+7)-2=0
thinking...
(-2+(2*abs((7+x)))+(abs((5+x))*x))=0
(((5+x)<0)&((((-2+(2*(7+x))+(-(5+x)*x))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0)))|(((-2+(2*-(7+x))+(-(5+x)*x))=0)&((7+x)<0))))|(((((-2+(2*(7+x))+((5+x)*x))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0)))|(((-2+(2*-(7+x))+((5+x)*x))=0)&((7+x)<0)))&((~((5+x)=0)&~((5+x)<0))|((5+x)=0)))
(((5+x)<0)&((((-2+(2*(-7-x))+((-5-x)*x))=0)&((7+x)<0))|(((-2+(2*(7+x))+((-5-x)*x))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0)))))|(((((-2+(2*(-7-x))+((5+x)*x))=0)&((7+x)<0))|(((-2+(2*(7+x))+((5+x)*x))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0))))&((~((5+x)=0)&~((5+x)<0))|((5+x)=0)))
(((5+x)<0)&((((-16-(7*x)-(x^2))=0)&((7+x)<0))|(((12-(3*x)-(x^2))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0)))))|(((((-16+(3*x)+(x^2))=0)&((7+x)<0))|(((12+(7*x)+(x^2))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0))))&((~((5+x)=0)&~((5+x)<0))|((5+x)=0)))
(((5+x)<0)&((((-16-(7*x)-(x^2))=0)&((7+x)<0))|((((((-3-sqrt(57))/2)-x)*(((3-sqrt(57))/2)+x))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0)))))|((((((3+x)*(4+x))=0)&((~((7+x)=0)&~((7+x)<0))|((7+x)=0)))|((((((3-sqrt(73))/2)+x)*(((3+sqrt(73))/2)+x))=0)&((7+x)<0)))&((~((5+x)=0)&~((5+x)<0))|((5+x)=0)))
{-3,-(3/2)-(sqrt(57)/2),-4}
=> {-3,-(3/2)-(sqrt(57)/2),-4}

? x*abs(x)-5*abs(x+2)+6=0
thinking...
(6+(abs(x)*x)-(5*abs((2+x))))=0
((x<0)&((((6+(-x*x)-(5*(-2-x)))=0)&((2+x)<0))|(((6+(-x*x)-(5*(2+x)))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))))|(((((6+(x*x)-(5*(-2-x)))=0)&((2+x)<0))|(((6+(x*x)-(5*(2+x)))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0))))&((~(x=0)&~(x<0))|(x=0)))
((x<0)&((((6+(5*(-2-x))-(x^2))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))|(((6+(5*(2+x))-(x^2))=0)&((2+x)<0))))|(((((6+(5*(-2-x))+(x^2))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))|(((6+(5*(2+x))+(x^2))=0)&((2+x)<0)))&((~(x=0)&~(x<0))|(x=0)))
((x<0)&((((-4-(5*x)-(x^2))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))|(((16+(5*x)-(x^2))=0)&((2+x)<0))))|(((((-4-(5*x)+(x^2))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))|(((16+(5*x)+(x^2))=0)&((2+x)<0)))&((~(x=0)&~(x<0))|(x=0)))
((x<0)&(((((-1-x)*(4+x))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))|((((((-5-sqrt(89))/2)+x)*(((5-sqrt(89))/2)-x))=0)&((2+x)<0))))|(((((16+(5*x)+(x^2))=0)&((2+x)<0))|((((((-5-sqrt(41))/2)+x)*(((-5+sqrt(41))/2)+x))=0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0))))&((~(x=0)&~(x<0))|(x=0)))
{(5/2)+(sqrt(41)/2),-1,(5/2)-(sqrt(89)/2)}
=> {(5/2)+(sqrt(41)/2),-1,(5/2)-(sqrt(89)/2)}

? x^2-abs(x+2)+x>0
thinking...
(-abs((2+x))+(x^2)+x)>0
((~((-(-2-x)+(x^2)+x)=0)&~((-(-2-x)+(x^2)+x)<0))&((2+x)<0))|((~((-(2+x)+(x^2)+x)=0)&~((-(2+x)+(x^2)+x)<0))&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))
(((2+x)<0)&~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|(~((-2+(x^2))=0)&~((-2+(x^2))<0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))
(((2+x)<0)&~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|(~(((((2*sqrt(2))/2)+x)*(-sqrt(2)+x))=0)&~(((((2*sqrt(2))/2)+x)*(-sqrt(2)+x))<0)&((~((2+x)=0)&~((2+x)<0))|((2+x)=0)))
(-inf,-sqrt(2))U((2*sqrt(2))/2,+inf)
=> (-inf,-sqrt(2))U((2*sqrt(2))/2,+inf)

? abs(3*x-5)+abs(8-x)=abs(3+2*x)
thinking...
(abs((-5+(3*x)))+abs((8-x))-abs((3+(2*x))))=0
(((3+(2*x))<0)&((((8-x)<0)&(((((-5+(3*x))-(-3-(2*x))-(8-x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|(((-(-3-(2*x))-(-5+(3*x))-(8-x))=0)&((-5+(3*x))<0))))|((((((-5+(3*x))+(8-x)-(-3-(2*x)))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|((((8-x)-(-3-(2*x))-(-5+(3*x)))=0)&((-5+(3*x))<0)))&((~((8-x)=0)&~((8-x)<0))|((8-x)=0)))))|(((((8-x)<0)&(((((-5+(3*x))-(3+(2*x))-(8-x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|(((-(-5+(3*x))-(3+(2*x))-(8-x))=0)&((-5+(3*x))<0))))|((((((-5+(3*x))+(8-x)-(3+(2*x)))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|((((8-x)-(-5+(3*x))-(3+(2*x)))=0)&((-5+(3*x))<0)))&((~((8-x)=0)&~((8-x)<0))|((8-x)=0))))&((~((3+(2*x))=0)&~((3+(2*x))<0))|((3+(2*x))=0)))
(((3+(2*x))<0)&((((8-x)<0)&(((0=0)&((-5+(3*x))<0))|(((-10+(6*x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))))|(((((16-(2*x))=0)&((-5+(3*x))<0))|(((6+(4*x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0))))&((~((8-x)=0)&~((8-x)<0))|((8-x)=0)))))|(((((8-x)<0)&((((-16+(2*x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|(((-6-(4*x))=0)&((-5+(3*x))<0))))|((((0=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|(((10-(6*x))=0)&((-5+(3*x))<0)))&((~((8-x)=0)&~((8-x)<0))|((8-x)=0))))&((~((3+(2*x))=0)&~((3+(2*x))<0))|((3+(2*x))=0)))
(((3+(2*x))<0)&((((8-x)<0)&((((-10+(6*x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|(((-5+(3*x))<0)&true)))|(((((16-(2*x))=0)&((-5+(3*x))<0))|(((6+(4*x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0))))&((~((8-x)=0)&~((8-x)<0))|((8-x)=0)))))|(((((8-x)<0)&((((-16+(2*x))=0)&((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0)))|(((-6-(4*x))=0)&((-5+(3*x))<0))))|(((((10-(6*x))=0)&((-5+(3*x))<0))|(((~((-5+(3*x))=0)&~((-5+(3*x))<0))|((-5+(3*x))=0))&true))&((~((8-x)=0)&~((8-x)<0))|((8-x)=0))))&((~((3+(2*x))=0)&~((3+(2*x))<0))|((3+(2*x))=0)))
(5/3,8)U{8,5/3}
=> (5/3,8)U{8,5/3}

? abs(x^2+5*x+9)<abs(x^2+2*x+2)+abs(3*x+7)
thinking...
(abs((9+(5*x)+(x^2)))-abs((2+(2*x)+(x^2)))-abs((7+(3*x))))<0
(((7+(3*x))<0)&((((2+(2*x)+(x^2))<0)&((((9+(5*x)+(x^2))<0)&((-(-2-(2*x)-(x^2))-(-7-(3*x))-(9+(5*x)+(x^2)))<0))|((((9+(5*x)+(x^2))-(-2-(2*x)-(x^2))-(-7-(3*x)))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0)))))|(((((9+(5*x)+(x^2))<0)&((-(-7-(3*x))-(2+(2*x)+(x^2))-(9+(5*x)+(x^2)))<0))|((((9+(5*x)+(x^2))-(-7-(3*x))-(2+(2*x)+(x^2)))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0))))&((~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|((2+(2*x)+(x^2))=0)))))|(((((2+(2*x)+(x^2))<0)&((((9+(5*x)+(x^2))<0)&((-(-2-(2*x)-(x^2))-(7+(3*x))-(9+(5*x)+(x^2)))<0))|((((9+(5*x)+(x^2))-(-2-(2*x)-(x^2))-(7+(3*x)))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0)))))|(((((9+(5*x)+(x^2))<0)&((-(2+(2*x)+(x^2))-(7+(3*x))-(9+(5*x)+(x^2)))<0))|((((9+(5*x)+(x^2))-(2+(2*x)+(x^2))-(7+(3*x)))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0))))&((~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|((2+(2*x)+(x^2))=0))))&((~((7+(3*x))=0)&~((7+(3*x))<0))|((7+(3*x))=0)))
(((7+(3*x))<0)&((((2+(2*x)+(x^2))<0)&(((0<0)&((9+(5*x)+(x^2))<0))|(((18+(10*x)+(2*(x^2)))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0)))))|(((((-4-(2*(x^2))-(4*x))<0)&((9+(5*x)+(x^2))<0))|(((14+(6*x))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0))))&((~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|((2+(2*x)+(x^2))=0)))))|(((((2+(2*x)+(x^2))<0)&((((-14-(6*x))<0)&((9+(5*x)+(x^2))<0))|(((4+(2*(x^2))+(4*x))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0)))))|((((0<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0)))|(((-18-(10*x)-(2*(x^2)))<0)&((9+(5*x)+(x^2))<0)))&((~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|((2+(2*x)+(x^2))=0))))&((~((7+(3*x))=0)&~((7+(3*x))<0))|((7+(3*x))=0)))
(((7+(3*x))<0)&((((2+(2*x)+(x^2))<0)&((((18+(10*x)+(2*(x^2)))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0)))|(((9+(5*x)+(x^2))<0)&false)))|(((((-4-(2*(x^2))-(4*x))<0)&((9+(5*x)+(x^2))<0))|(((14+(6*x))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0))))&((~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|((2+(2*x)+(x^2))=0)))))|(((((2+(2*x)+(x^2))<0)&((((-14-(6*x))<0)&((9+(5*x)+(x^2))<0))|(((4+(2*(x^2))+(4*x))<0)&((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0)))))|(((((-18-(10*x)-(2*(x^2)))<0)&((9+(5*x)+(x^2))<0))|(((~((9+(5*x)+(x^2))=0)&~((9+(5*x)+(x^2))<0))|((9+(5*x)+(x^2))=0))&false))&((~((2+(2*x)+(x^2))=0)&~((2+(2*x)+(x^2))<0))|((2+(2*x)+(x^2))=0))))&((~((7+(3*x))=0)&~((7+(3*x))<0))|((7+(3*x))=0)))
(-inf,-(7/3))
=> (-inf,-(7/3))

? dif(y,x)=sqrt(4-y^2)
thinking...
(-arcsin((y/2))+x+c1)=0
=> (-arcsin((y/2))+x+c1)=0

? dif(y,x)+y=1
thinking...
(log(abs((1-y)))+x+c1)=0
=> (log(abs((1-y)))+x+c1)=0

? x^5*dif(y,x)=-y^5
thinking...
((4*(x^4)*(y^4)*c1)+(x^4)+(y^4))=0
=> ((4*(x^4)*(y^4)*c1)+(x^4)+(y^4))=0

? dif(y,x)=(1+x^2)*(1+y^2)
thinking...
((3*x)+(3*c1)-(3*arctan(y))+(x^3))=0
=> ((3*x)+(3*c1)-(3*arctan(y))+(x^3))=0

? x*(x^2-1)*dif(y,x)=1
thinking...
(integrate((1/((-1+x)*(1+x)*x)),x)-y+c1)=0
((2*log(abs(x)))+(2*c1)-log(abs((-1+x)))-log(abs((1+x)))-(2*y))=0
=> ((2*log(abs(x)))+(2*c1)-log(abs((-1+x)))-log(abs((1+x)))-(2*y))=0

? dif(y,x)=arcsin(x)
thinking...
(integrate(arcsin(x),x)-y+c1)=0
(-y+try(((arcsin(x)*x)+sqrt((1-(x^2)))),integrate(arcsin(x),x))+c1)=0
=> ((arcsin(x)*x)-y+sqrt((1-(x^2)))+c1)=0

? (x^2+x*y)*dif(y,x)=(x^2+y^2)
thinking...
try(subs((integrate(((1+z)/(-1+z)),z)+log(abs(x))+c1),z,(y/x)))=0
try(subs((integrate((z/(-1+z)),z)+log(abs((-1+z)))+log(abs(x))+c1),z,(y/x)))=0
try(subs((log(abs((-1+z)))+log(abs(x))+try(subs(integrate(((1+a)/a),a),a,(-1+z)),subs((-(log(abs((a^2)))/2)-integrate((1/(a^2)),a)),a,(1/(-1+z))))+c1),z,(y/x)))=0
try(subs((log(abs((-1+z)))+log(abs(x))+try(subs((log(abs(a))+a),a,(-1+z)),subs(((2-(log(abs((a^2)))*a))/(2*a)),a,(1/(-1+z))))+c1),z,(y/x)))=0
=> ((2*log(abs(((-x+y)/x)))*x)+(log(abs(x))*x)+(x*c1)-x+y)=0

? dif(y,x)=(x+y)/x
thinking...
try(subs((log(abs(x))-z+c1),z,(y/x)))=0
=> ((log(abs(x))*x)+(x*c1)-y)=0

? (x-y)*dif(y,x)-(x+y)=0
thinking...
try(subs(((log(abs((-1-(z^2))))/2)+log(abs(x))-arctan(z)+c1),z,(y/x)))=0
=> (log(abs(((-(x^2)-(y^2))/(x^2))))+(2*log(abs(x)))+(2*c1)-(2*arctan((y/x))))=0

? (x^2-y^2)+2*x*y*dif(y,x)=0
thinking...
try(subs((((2*c1)-log(abs((1+(z^2))))-log(abs(x)))/2),z,(y/x)))=0
=> ((2*c1)-log(abs((1+((y/x)^2))))-log(abs(x)))=0

? x*dif(y,x)-y+x*sin(y/x)=0
thinking...
try(subs((-log(abs(tan((z/2))))-log(abs(x))+c1),z,(y/x)))=0
=> (-log(abs((sin((y/(2*x)))/cos((y/(2*x))))))-log(abs(x))+c1)=0

? (x+y)*dif(y,x)=1
thinking...
(((e^-y)*x)-integrate(((e^-y)*y),y)-c1)=0
(((e^-y)*x)-try(subs(integrate(log(z),z),z,(e^-y)),subs(integrate(((e^z)*z),z),z,-y))-c1)=0
(((e^-y)*x)-try(subs(integrate(log(z),z),z,(e^-y)),subs(subs(integrate(log(a),a),a,(e^z)),z,-y))-c1)=0
(((e^-y)*x)-try(subs(try(((log(z)*z)-z),integrate(log(z),z)),z,(e^-y)),subs(subs(try(((log(a)*a)-a),integrate(log(a),a)),a,(e^z)),z,-y))-c1)=0
=> (((e^-y)*x)-(log((e^-y))*(e^-y))-c1+(e^-y))=0

? x^2*dif(y,x)=x^2-2*y^2+x*y
thinking...
try(subs((-log(abs(x))-(8*integrate((1/(((2*sqrt(2))+(4*z))*((4*z)-(2*sqrt(2))))),z))+c1),z,(y/x)))=0
try(subs(((2*(-(16*integrate((1/(-1+(4*z)-(8*sqrt(2)*x))),z)*x)-(log(abs(((4*z)-(2*sqrt(2)))))*x)))-log(abs(x))+c1),z,(y/x)))=0
try(subs((-log(abs(x))-(2*log(abs(((4*z)-(2*sqrt(2)))))*x)-(32*integrate((1/(-1+(4*z)-(8*sqrt(2)*x))),z)*x)+c1),z,(y/x)))=0
try(subs((-log(abs(x))-(2*log(abs(((4*z)-(2*sqrt(2)))))*x)-(32*try(subs((log(abs(a))/4),a,(-1+(4*z)-(8*sqrt(2)*x))),subs(-(log(abs(a))/4),a,(1/(-1+(4*z)-(8*sqrt(2)*x)))))*x)+c1),z,(y/x)))=0
=> (-log(abs(x))-(2*log(abs((((4*y)-(2*sqrt(2)*x))/x)))*x)-(8*log(abs((((4*y)-(8*(x^2)*sqrt(2))-x)/x)))*x)+c1)=0

? integrate(1/(1+sin(x)),x)
thinking...
integrate((1/(1+sin(x))),x)
integrate(((1-sin(x))/(cos(x)^2)),x)
-integrate((sin(x)/(cos(x)^2)),x)+tan(x)
(-(2*cos(x)*integrate((sin(x)/(1+cos((2*x)))),x))+sin(x))/cos(x)
(-(2*cos(x)*try(subs((integrate((sin((y/2))/(1+cos(y))),y)/2),y,(2*x)),subs(integrate((y/((1+cos((2*arcsin(y))))*cos(arcsin(y)))),y),y,sin(x)),subs(-(integrate((sin((arccos(y)/2))/((1+y)*sin(arccos(y)))),y)/2),y,cos((2*x))),subs(-(integrate((sin((arccos((-1+y))/2))/(sin(arccos((-1+y)))*y)),y)/2),y,(1+cos((2*x)))),subs((integrate((sin((arccos(((1-y)/y))/2))/(sin(arccos(((1-y)/y)))*y)),y)/2),y,(1/(1+cos((2*x)))))))+sin(x))/cos(x)
(-(2*cos(x)*try((((2*cos((2*x))*integrate(((cos(x)*sin((2*x)))/((1+cos((2*x)))^2)),x))+(2*integrate(((cos(x)*sin((2*x)))/((1+cos((2*x)))^2)),x))-cos(x))/(1+cos((2*x)))),((integrate((1/(1+cos((2*x)))),x)*sin(x))-integrate((cos(x)*integrate((1/(1+cos((2*x)))),x)),x))))+sin(x))/cos(x)
-try(subs(integrate((y/(cos(arcsin(y))^3)),y),y,sin(x)),subs((1/y),y,cos(x)),subs((integrate(cos((1/sqrt(y))),y)/2),y,(1/(cos(x)^2))),subs(-(integrate((1/(cos(sqrt(y))^3)),y)/2),y,(cos(x)^2)))+tan(x)
=> (-1+sin(x))/cos(x)

? integrate((7^(7^(7^x)))*(7^(7^x))*(7^x),x)
thinking...
integrate((7^((7^(7^x))+(7^x)+x)),x)
subs(((7^y)/(log(7)^3)),y,(7^(7^x)))
=> (7^(7^(7^x)))/(log(7)^3)

? integrate(abs(x+1),x,-4,10)
thinking...
-subs((((2*x)+(x^2))/2),x,-1)-subs(((-(2*x)-(x^2))/2),x,-4)+subs((((2*x)+(x^2))/2),x,10)+subs(((-(2*x)-(x^2))/2),x,-1)
=> 65

? integrate(abs(sin(x)),x,-pi,pi/6)
thinking...
-subs(cos(x),x,-pi)-subs(-cos(x),x,0)+subs(cos(x),x,0)+subs(-cos(x),x,(pi/6))
=> (6-sqrt(3))/2

? integrate(2*x/(1+x^2),x)
thinking...
log(abs((1+(x^2))))
=> log(abs((1+(x^2))))

? integrate(sqrt(a*x+b),x)
thinking...
(2*(((x*a)+b)^(3/2)))/(3*a)
=> (2*(((x*a)+b)^(3/2)))/(3*a)

? integrate(x*sqrt(x),x)
thinking...
(2*(x^(5/2)))/5
=> (2*(x^(5/2)))/5

? integrate(x*sqrt(1+2*x^2),x)
thinking...
sqrt((1+(2*(x^2))))/2
=> sqrt((1+(2*(x^2))))/2

? integrate(x/sqrt(x+4),x)
thinking...
integrate(((1/sqrt((4+x)))*x),x)
try(subs(integrate(((-4+y)*(1/sqrt(y))),y),y,(4+x)),subs((2*integrate(((-1+(4*(y^2)))/(y^4)),y)),y,(1/sqrt((4+x)))))
try(subs(((2*(-sqrt((144*y))+(y^(3/2))))/3),y,(4+x)),subs((2*integrate(((-1+(4*(y^2)))/(y^4)),y)),y,(1/sqrt((4+x)))))
=> (2*(-sqrt((144*(4+x)))+((4+x)^(3/2))))/3

? integrate(e^(2*x+3),x)
thinking...
(e^(3+(2*x)))/2
=> (e^(3+(2*x)))/2

? integrate(x/e^(x^2),x)
thinking...
integrate(((e^-(x^2))*x),x)
try(subs(-((e^-y)/2),y,(x^2)),subs(-(y/2),y,(e^-(x^2))))
=> -((e^-(x^2))/2)

? integrate(sqrt(sin(2*x))*cos(2*x),x)
thinking...
integrate((cos((2*x))*sqrt(sin((2*x)))),x)
integrate((cos((2*x))*(1/sqrt(sin((2*x))))),x)
try(subs(sqrt(y),y,sin((2*x))),subs(-integrate(sin((1/(y^2))),y),y,(1/sqrt(sin((2*x))))),subs(-(integrate(((sin(arccos(y))^-(3/2))*y),y)/2),y,cos((2*x))),subs((integrate((cos(y)*(1/sqrt(sin(y)))),y)/2),y,(2*x)))
=> sqrt(sin((2*x)))

? integrate(sin(x)*sin(cos(x)),x)
thinking...
integrate((sin(cos(x))*sin(x)),x)
try(subs(-integrate((sin(cos(arcsin(y)))/cos(cos(arcsin(y)))),y),y,sin(cos(x))),subs(integrate(((sin(cos(arcsin(y)))*y)/cos(arcsin(y))),y),y,sin(x)),subs(cos(y),y,cos(x)))
=> cos(cos(x))

? integrate(sin(2*x+5)^2,x)
thinking...
integrate((sin((5+(2*x)))^2),x)
((4*x)-sin((10+(4*x))))/8
=> ((4*x)-sin((10+(4*x))))/8

? integrate(sin(3*x)*cos(4*x),x)
thinking...
integrate((cos((4*x))*sin((3*x))),x)
((7*cos(x))-cos((7*x)))/14
=> ((7*cos(x))-cos((7*x)))/14

? integrate(cos(2*x)*cos(4*x)*cos(6*x),x)
thinking...
integrate((cos((2*x))*cos((4*x))*cos((6*x))),x)
((2*sin((12*x)))+(24*x)+(3*sin((8*x)))+(6*sin((4*x))))/96
=> ((2*sin((12*x)))+(24*x)+(3*sin((8*x)))+(6*sin((4*x))))/96

? integrate(sin(2*x+1)^3,x)
thinking...
integrate((sin((1+(2*x)))^3),x)
(cos((3+(6*x)))-(3*cos((-1-(2*x))))-(6*cos((1+(2*x)))))/24
=> (cos((3+(6*x)))-(3*cos((-1-(2*x))))-(6*cos((1+(2*x)))))/24

? integrate(sin(x)^3*cos(x)^3,x)
thinking...
integrate(((cos(x)^3)*(sin(x)^3)),x)
(cos((6*x))-(9*cos((2*x))))/192
=> (cos((6*x))-(9*cos((2*x))))/192

? integrate(sin(x)*sin(2*x)*sin(3*x),x)
thinking...
integrate((sin((2*x))*sin((3*x))*sin(x)),x)
((2*cos((6*x)))-(3*cos((4*x)))-(6*cos((2*x))))/48
=> ((2*cos((6*x)))-(3*cos((4*x)))-(6*cos((2*x))))/48

? integrate(sin(4*x)*sin(8*x),x)
thinking...
integrate((sin((4*x))*sin((8*x))),x)
((3*sin((4*x)))-sin((12*x)))/24
=> ((3*sin((4*x)))-sin((12*x)))/24

? integrate(sin(x)^4,x)
thinking...
integrate((sin(x)^4),x)
((12*x)-(8*sin((2*x)))+sin((4*x)))/32
=> ((12*x)-(8*sin((2*x)))+sin((4*x)))/32

? integrate(cos(2*x)^4,x)
thinking...
integrate((cos((2*x))^4),x)
((24*x)+(8*sin((4*x)))+sin((8*x)))/64
=> ((24*x)+(8*sin((4*x)))+sin((8*x)))/64

? integrate(x/((x+1)*(x+2)),x)
thinking...
(log(abs((2+(3*x)+(x^2))))/2)-((3*integrate((1/(2+(3*x)+(x^2))),x))/2)
(log(abs((2+(3*x)+(x^2))))+(3*log(abs((1+x))))-(3*log(abs((2+x)))))/2
=> (log(abs((2+(3*x)+(x^2))))+(3*log(abs((1+x))))-(3*log(abs((2+x)))))/2

? integrate(1/(x^2-9),x)
thinking...
integrate((1/((-3+x)*(3+x))),x)
(log(abs((3+x)))-log(abs((-3+x))))/6
=> (log(abs((3+x)))-log(abs((-3+x))))/6

? integrate((3*x-1)/((x-1)*(x-2)*(x-3)),x)
thinking...
integrate(((-1+(3*x))/((-1+x)*(-2+x)*(-3+x))),x)
(3*integrate((x/((-1+x)*(-2+x)*(-3+x))),x))-integrate((1/((-1+x)*(-2+x)*(-3+x))),x)
(5*log(abs((-2+x))))-log(abs((-1+x)))-(4*log(abs((-3+x))))
=> (5*log(abs((-2+x))))-log(abs((-1+x)))-(4*log(abs((-3+x))))

? integrate(x/((x-1)*(x-2)*(x-3)),x)
thinking...
integrate((x/((-1+x)*(-2+x)*(-3+x))),x)
((4*log(abs((-2+x))))-log(abs((-1+x)))-(3*log(abs((-3+x)))))/2
=> ((4*log(abs((-2+x))))-log(abs((-1+x)))-(3*log(abs((-3+x)))))/2

? integrate(2*x/(x^2+3*x+2),x)
thinking...
2*((log(abs((2+(3*x)+(x^2))))/2)-((3*integrate((1/(2+(3*x)+(x^2))),x))/2))
log(abs((2+(3*x)+(x^2))))+(3*log(abs((1+x))))-(3*log(abs((2+x))))
=> log(abs((2+(3*x)+(x^2))))+(3*log(abs((1+x))))-(3*log(abs((2+x))))

? integrate((1-x^2)/(x*(1-2*x)),x)
thinking...
integrate((((1-x)*(1+x))/((1-(2*x))*x)),x)
integrate((1/((1-(2*x))*x)),x)-integrate((x/(1-(2*x))),x)
log(abs((1-(2*x))))-integrate((x/(1-(2*x))),x)-log(abs(x))
log(abs((1-(2*x))))-log(abs(x))-try(subs(((log(abs((y^2)))-(2*integrate((1/(y^2)),y)))/8),y,(1/(1-(2*x)))),subs((integrate(((2*(-1+y))/y),y)/8),y,(1-(2*x))))
log(abs((1-(2*x))))-log(abs(x))-try(subs(((2+(log(abs((y^2)))*y))/(8*y)),y,(1/(1-(2*x)))),subs(((-log(abs(y))+y)/4),y,(1-(2*x))))
=> (-2+(10*log(abs((1/((1-(2*x))^2))))*x)+(24*x)+(256*log(abs(x))*(x^5))+(32*log(abs((1/((1-(2*x))^2))))*(x^5))+(320*log(abs((1-(2*x))))*(x^2))+(320*(x^3))+(384*(x^5))+(640*log(abs((1-(2*x))))*(x^4))+(640*log(abs(x))*(x^3))+(8*log(abs((1-(2*x)))))+(80*log(abs((1/((1-(2*x))^2))))*(x^3))+(80*log(abs(x))*x)-log(abs((1/((1-(2*x))^2))))-(120*(x^2))-(128*(x^6))-(256*log(abs((1-(2*x))))*(x^5))-(320*log(abs(x))*(x^2))-(40*log(abs((1/((1-(2*x))^2))))*(x^2))-(480*(x^4))-(640*log(abs((1-(2*x))))*(x^3))-(640*log(abs(x))*(x^4))-(8*log(abs(x)))-(80*log(abs((1-(2*x))))*x)-(80*log(abs((1/((1-(2*x))^2))))*(x^4)))/(8*((1-(2*x))^5))

? integrate(x/((x-1)*(x^2+1)),x)
thinking...
integrate((x/((-1+x)*(1+(x^2)))),x)
(log(abs((1+(x^2))))-(2*arctan(x))-(2*log(abs((-1+x)))))/4
=> (log(abs((1+(x^2))))-(2*arctan(x))-(2*log(abs((-1+x)))))/4

? integrate(x/((x-1)^2*(x+2)),x)
thinking...
integrate((x/((2+x)*((-1+x)^2))),x)
(3+(2*log(abs((-1+x))))+(2*log(abs((2+x)))*x)-(2*log(abs((-1+x)))*x)-(2*log(abs((2+x)))))/(9*(-1+x))
=> (3+(2*log(abs((-1+x))))+(2*log(abs((2+x)))*x)-(2*log(abs((-1+x)))*x)-(2*log(abs((2+x)))))/(9*(-1+x))

? integrate((2*x-3)/((x^2-1)*(2*x+3)),x)
thinking...
integrate(((-3+(2*x))/((-1+x)*(1+x)*(3+(2*x)))),x)
(2*integrate((x/((-1+x)*(1+x)*(3+(2*x)))),x))-(3*integrate((1/((-1+x)*(1+x)*(3+(2*x)))),x))
(log(abs((-1+x)))+(24*log(abs((3+(2*x)))))-(25*log(abs((1+x)))))/10
=> (log(abs((-1+x)))+(24*log(abs((3+(2*x)))))-(25*log(abs((1+x)))))/10

? integrate(5*x/((x+1)*(x^2-4)),x)
thinking...
5*integrate((x/((-2+x)*(1+x)*(2+x))),x)
(5*((3*log(abs((2+x))))-log(abs((-2+x)))-(2*log(abs((1+x))))))/6
=> (5*((3*log(abs((2+x))))-log(abs((-2+x)))-(2*log(abs((1+x))))))/6

? integrate(2/((1-x)*(1+x^2)),x)
thinking...
2*integrate((1/((1-x)*(1+(x^2)))),x)
((2*log(abs((1-x))))-log(abs((1+(x^2))))-(2*arctan(x)))/2
=> ((2*log(abs((1-x))))-log(abs((1+(x^2))))-(2*arctan(x)))/2

List of Questions from NCERT

Class 12 – Differential Equations (Chapter 9, Exercise 9.3)

1. dif(y,x)=sqrt(4-y^2)
2. dif(y,x)+y=1
3. x^5*dif(y,x)=-y^5

Class 12 – Differential Equations (Chapter 9, Exercise 9.4)

4. (x^2+x*y)*dif(y,x)=(x^2+y^2)
5. dif(y,x)=(x+y)/x
6. (x-y)*dif(y,x)-(x+y)=0
7. (x^2-y^2)+2xy*dif(y,x)=0
8. xdif(y,x)-y+xsin(y/x)=0

Class 12 – Differential Equations (Chapter 9, Exercise 9.5)

9. (x+y)*dif(y,x)=1
10. x^2dif(y,x)=x^2-2y^2+x*y

Class 12 – Integrals (Chapter 7, Exercise 7.2)

11. integrate(2*x/(1+x^2),x)
12. integrate(sqrt(a*x+b),x)
13. integrate(x*sqrt(x),x)
14. integrate(xsqrt(1+2x^2),x)
15. integrate(x/sqrt(x+4),x)
16. integrate(e^(2*x+3),x)
17. integrate(x/e^(x^2),x)
18. integrate(sqrt(sin(2*x))cos(2x),x)
19. integrate(sin(x)*sin(cos(x)),x)

Class 12 – Integrals (Chapter 7, Exercise 7.3)

20. integrate(sin(2*x+5)^2,x)
21. integrate(sin(3*x)cos(4x),x)
22. integrate(cos(2*x)cos(4x)cos(6x),x)
23. integrate(sin(2*x+1)^3,x)
24. integrate(sin(x)^3*cos(x)^3,x)
25. integrate(sin(x)sin(2x)sin(3x),x)
26. integrate(sin(4*x)sin(8x),x)
27. integrate(sin(x)^4,x)
28. integrate(cos(2*x)^4,x)

Class 12 – Integrals (Chapter 7, Exercise 7.5)

29. integrate(x/((x+1)*(x+2)),x)
30. integrate(1/(x^2-9),x)
31. integrate((3x-1)/((x-1)(x-2)*(x-3)),x)
32. integrate(x/((x-1)(x-2)(x-3)),x)
33. integrate(2x/(x^2+3x+2),x)
34. integrate((1-x^2)/(x*(1-2*x)),x)
35. integrate(x/((x-1)*(x^2+1)),x)
36. integrate(x/((x-1)^2*(x+2)),x)
37. integrate((2x-3)/((x^2-1)(2*x+3)),x)
38. integrate(5x/((x+1)(x^2-4)),x)
39. integrate(2/((1-x)*(1+x^2)),x)

Class 10 – Trigonometry (Chapter 8, Exercise 8.3)

40. (cosec(x)-cot(x))^2=(1-cos(x))/(1+cos(x))
41. cos(x)/(1+sin(x))+(1+sin(x))/cos(x)=2*sec(x)
42. tan(x)/(1-cot(x))+cot(x)/(1-tan(x))=1+sec(x)*cosec(x)
43. (1+sec(x))/sec(x)=sin(x)^2/(1-cos(x))
44. (cos(x)-sin(x)+1)/(cos(x)+sin(x)-1)=cosec(x)+cot(x)
45. (sin(x)-2sin(x)^3)/(2cos(x)^3-cos(x))=tan(x)
46. (sin(x)+cosec(x))^2+(cos(x)+sec(x))^2=7+tan(x)^2+cot(x)^2
47. (cosec(x)-sin(x))*(sec(x)-cos(x))=1/(tan(x)+cot(x))

Class 11 – Inequalities (Chapter 5, Exercise 5.1)

48. 4x+3<5x+7
49. x/2>=(5x-2)/3-(7x-3)/5

Class 11 – Sets (Chapter 1, Miscellaneous)

50. (A-B)|(B-A)<->(A|B)-(A&B)
51. A|(B-A)<->(A|B)