calculus - How can we tackle this integral $int_{0}^{1}{2x^2-2x+ln[(1-x)(1+x)^3]over x^3sqrt{1-x^2}}mathrm dx=-1?$

Something is wrong with this integral (in terms of splitting them out)

$$\int_{0}^{1}{2x^2-2x+\ln[(1-x)(1+x)^3]\over x^3\sqrt{1-x^2}}\mathrm dx=\color{blue}{-1}\tag1$$

My try:

Splitting the integral

$$\int_{0}^{1}{2x^2-2x\over x^3\sqrt{1-x^2}}\mathrm dx+\int_{0}^{1}{\ln[(1-x)(1+x)^3]\over x^3\sqrt{1-x^2}}\mathrm dx=I_1+I_2\tag2$$

Note that $I_1$ and $I_2$ diverge, so how can we tackle it as a whole?

Answer

I will present an answer without Gamma functions and without series' expansions. Instead, a more general problem is solved, where the OP's question is a special case.

Let
$$I(a) = \int_{0}^{1}{2a^2x^2-2ax+\ln[(1-ax)(1+ax)^3]\over x^3\sqrt{1-x^2}}\mathrm dx$$

The answer to the OP's question is then given by $I(a = 1)$.

Then, after partial differentiation w.r.t. $a$,

$$I'(a) = \int_{0}^{1} \frac{2a^2(2ax - 1)}{(a^2 x^2 - 1)\sqrt{1-x^2}}\mathrm dx$$

Note that this removes the problematic factor $x^3$ in the denominator - this issue was solved in the previous solution by series' expansion of the $\log$. Hence, convergence is established and the order of integrations $x,a$ can be exchanged.

The $x$-integration gives
$$I'(a) = a^2 \Big[ \frac{2 \arctan(\frac{x \sqrt{1 - a^2}}{\sqrt{1 - x^2}})}{\sqrt{1 - a^2}} + \frac{4\arctan(\frac{a \sqrt{1 - x^2}}{\sqrt{1 - a^2}})}{\sqrt{1 - a^2}} \Big]_{x=0}^{1}$$

which is

$$I'(a) = a^2 \Big[ \frac{\pi}{\sqrt{1 - a^2}} - \frac{4\arctan(\frac{a }{\sqrt{1 - a^2}})}{\sqrt{1 - a^2}} \Big] = a^2 \Big[ \frac{\pi}{\sqrt{1 - a^2}} - \frac{4\arcsin({a })}{\sqrt{1 - a^2}} \Big]$$

Now we integrate w.r.t. $a$:
$$I(a) = \int I'(a) \rm{d} a = \int a^2 \Big[ \frac{\pi}{\sqrt{1 - a^2}} - \frac{4\arcsin({a })}{\sqrt{1 - a^2}} \Big] \rm{d} a$$

Replacing $a = \sin (y)$ gives

$$I(y) = \int \sin^2(y) \Big[ {\pi} - {4 y}\Big] \rm{d} y = y \sin(2y) + (\pi y)/2 - \sin^2(y) - y^2 - (\pi \sin(2y))/4 + C$$

We need the constant $C$. Obviously $0 = I(a=0) = I(y=0)$ which gives $C=0$.

The function in question by the OP is $I(a=1) = I(y=\pi/2) = -1$.

This solves the OP'S question. $\qquad \qquad \Box$