Real method solving $\int_{0}^{\infty} x^3 K_0(x)^5I_0(x)\left ( \pi^2I_0(x)^2-K_0(x)^2 \right ) \text{d}x=\frac{\pi^4}{128}$

Question

Define two zero order Bessel functions: $$ K_0(x)=\int_{0}^{\infty}e^{-x\cosh(t)}\text{d}t\quad \text{for}\operatorname{Re}(x)>0 $$ $$ I_0(x)=\frac1\pi\int_{0}^{\pi}e^{x\cos(t)}\text{d}t $$ By considering the function $$f(z)=z^3{K_0(z)}^4{K_0(-z)}^4$$, it is known that $$\int_{0}^{\infty} x^3 K_0(x)^5I_0(x)\left ( \pi^2I_0(x)^2-K_0(x)^2 \right ) \text{d}x=\frac{\pi^4}{128}.$$

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My question would be: Can we do that one in real methods? Just for curiosity. I have tried several integral transformations but failed. Multiple integration also suggests no result. But there's still a possibility that I don't get on the right way. So I need your cures.

Answer 1

I have found it. Take a look at here.