How Do You Integrate 1/(x^2 + 1)^2 Using the Hermite-Ostrogradski Method?

[righteous818]

i am have been at this whole day can you tell me how to integrate 1/(x^2 +1)^2

 

[CaptainBlack]

i am have been at this whole day can you tell me how to integrate 1/(x^2 +1)^2

integral 1/(x^2+1)^2 - Wolfram|Alpha

and click on show steps.

CB

 

[Fernando Revilla]

i am have been at this whole day can you tell me how to integrate 1/(x^2 +1)^2

An alternative: the problem is just routine if you know the Hermite-Ostrogradski method.

Denoting $q(x)=(x^2+1)^2$ we have

$q_1(x)=\gcd \left\{ q(x),q'(x)\right\}=\gcd \left\{ (x^2+1)^2,2x(x^2+1) \right\}=x^2+1$

$q_2(x)=\dfrac{q(x)}{q_1(x)}=x^2+1$

Then,

$\displaystyle\int \dfrac{1}{(x^2+1)^2}\;dx=\dfrac{Ax+B}{q_1(x)}+\int \dfrac{Cx+D}{q_2(x)}\;dx$

equivalently:

$\displaystyle\int \dfrac{1}{(x^2+1)^2}\;dx=\dfrac{Ax+B}{x^2+1}+\int \dfrac{Cx+D}{x^2+1}\;dx$

and we can determine $A,B,C,D$ differentiating both sides with respect to x.