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I've been attempting to integrate the following:

$$\int \frac{d^3 p}{(2\pi)^3} \frac{p}{T} \sin^2 (2\theta) \frac{1}{e^{p/T}+1}$$

to do this I chose to use spherical coordinates which originated the problem mentioned in How do I integrate this by parts? (if it is by parts)

The use of such coordinates leads to this equation:

$$\frac{1}{2\pi^3}\int^{2\pi}_0 \int^ \pi _0 \int ^\infty _0 dp d\theta d\phi \hspace{1mm}\frac{p}{T} \sin^2 2\theta \frac{1}{e^{p/T} +1} p^2 \sin \theta $$

I have attempted integrating it but been unsuccessful and that link is one of the reasons as to why.

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Hint:$$\int_0^\pi\sin\theta\sin^22\theta d\theta=8\int_0^{\pi/2}\sin^3\theta\cos^2\theta d\theta=4\operatorname{B}(2,\,\tfrac32)=\frac{16}{15}$$and$$\int_0^\infty\frac{x^3dx}{e^x+1}=\Gamma(4)\eta(4)=24(1-2^{-3})\zeta(4)=\frac{7\pi^4}{30}.$$

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