<s>[INST] computing integral without softwares: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$ I was wondering if this integral can be solve without wolfram and others:
$$\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$$
Thanks.
 [/INST] This is a plan of attack, at least:
1) Separate into two terms, and to find $\displaystyle\int\frac{2x}{x^2+\sqrt{1-x^2}} dx$, let $u=x^2$ to get $\displaystyle\int\frac{1}{u+\sqrt{1-u}} du$, and then let $t=\sqrt{1-u}$ to obtain $$ 2\int\frac{t}{t^2-t-1} dt.$$
Now use partial fractions.
2) For the second term, let $x=\sin\theta$ to get $\displaystyle3\int\frac{1}{x^2+\sqrt{1-x^2}} dx=3\int\frac{\cos\theta}{\sin^{2}\theta+\cos\theta} d\theta$.
Next let $\theta=\tan\frac{t}{2}, \sin\theta=\frac{2t}{1+t^2},  \cos\theta=\frac{1-t^2}{1+t^2}, d\theta=\frac{2}{1+t^2} dt$ to get
 $$\int\frac{2(1-t^2)}{4t^2+(1+t^2)(1-t^2)} dt=2\int\frac{t^2-1}{t^4-4t^2-1} dt.$$
Now use partial fractions.
</s>