1033.sagews

integral(x^2 *exp(x), x)

(x^2 - 2*x + 2)*e^x

i1=integral(x^2 *exp(x), x)

i1

(x^2 - 2*x + 2)*e^x

diff( i1, x)

2*(x - 1)*e^x + (x^2 - 2*x + 2)*e^x

expand( diff(i1,x))

x^2*e^x

integral(arctan(x) , x)

x*arctan(x) - 1/2*log(x^2 + 1)

diff( _ , x)

arctan(x)

r=x^3/(1+x^2)

r.partial_fraction(x)

x - x/(x^2 + 1)

(x^3/(1+x^2)).partial_fraction(x)

x - x/(x^2 + 1)

integral(x^3/(1+x^2),x)

1/2*x^2 - 1/2*log(x^2 + 1)

(x^3/(x^2-2*x+1)).partial_fraction(x)

x + 3/(x - 1) + 1/(x - 1)^2 + 2

integral( x^3/(x^2-2*x+1),x)

-1/(x - 1) + 1/2*x^2 + 2*x + 3*log(x - 1)

diff( _ ,x)

x + 3/(x - 1) + 1/(x - 1)^2 + 2

factor(_)

x^3/(x - 1)^2