I was solving another problem, and it remains to prove the following integral: $$ \int_0^{\infty }\left(3{\exp \left(-2t \right)\over t}\operatorname{erf}^2(\sqrt t) -\frac{\sqrt\pi}2{\exp \left(-t \right)\over t^{3/2}}\operatorname{erf}^3(\sqrt t)\right)dt=\frac{\pi}4 $$ Can somebody give me a hint?
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$$\left(\frac\pi4\text{erf}\left(\sqrt{t}\right)^4+\sqrt{\pi\over t}e^{-t} \text{erf}\left(\sqrt{t}\right)^3\right)'=3{\exp \left(-2t \right)\over t}\operatorname{erf}^2(\sqrt t) -\frac{\sqrt\pi}2{\exp \left(-t \right)\over t^{3/2}}\operatorname{erf}^3(\sqrt t)$$ Evaluating the limit $$\lim_{t\to0}\frac\pi4\text{erf}\left(\sqrt{t}\right)^4+\sqrt{\pi\over t}e^{-t} \text{erf}\left(\sqrt{t}\right)^3=0+0=0$$ and$$\lim_{t\to\infty}\frac\pi4\text{erf}\left(\sqrt{t}\right)^4+\sqrt{\pi\over t}e^{-t} \text{erf}\left(\sqrt{t}\right)^3=\frac\pi4\times1^4+0\times1^3=\frac\pi4$$ So the integral is $\pi\over4$