<s>[INST] Can $(x^2+a^2)^{-\frac{3}{2}}$ be integrated without using trigonometric substitutions? I know that
$$
\int\frac{1}{(x^2+a^2)^{\frac{3}{2}}}\,dx
$$
can be solved using a trigonometric substitution, but is there a trick to calculate this integral without using trigonometric substitutions? I think something in the same spirit with solving
$$
\int\frac{1}{(x^2+1)^2}\,dx
$$
by adding and subtracting $x^2$ to the numerator and then integrating by parts once one knows (and it is simple) how to integrate
$$
\int\frac{1}{(x^2+1)}\,dx
$$
Any suggestion or reference will be appreciated.
 [/INST] A very simple approach would be:-
$$\int\frac{1}{(x^2+a^2)^{\frac{3}{2}}}\,dx=\int\frac{1}{(x^3)(1+\frac{a^2}{x^2})^{\frac{3}{2}}}\,dx$$
Substituting $(1+\frac{a^2}{x^2})$ as $t^2$,we get :
$$\int\frac{-1}{a^2t^2}dt=\frac{1}{a^2t} +c=\frac{x}{(a^2){(x^2+a^2)^{\frac{1}{2}}}} +c  $$
</s>