<s>[INST] What is the indefinite integral of $\sqrt[4] {\tan \left( x\right) }$ $$\int\sqrt[4] {\tan \left( x\right) } dx$$
I'm really stuck right now with this integral, so any kind of advice would be appreciated. 
Perhaps a pretty nifty substitution that can save me from partial fraction decomposition.
 [/INST] [Too long for a comment]
Mathematica gives this answer. 
$\displaystyle\int \sqrt[4]{\tan (x)} \, dx=\\\cos \left(\frac{\pi }{8}\right) \tan ^{-1}\left(\csc \left(\frac{\pi }{8}\right) \left(\sqrt[4]{\tan (x)}-\cos \left(\frac{\pi }{8}\right)\right)\right)+\\\cos \left(\frac{\pi }{8}\right) \tan ^{-1}\left(\csc \left(\frac{\pi }{8}\right) \left(\sqrt[4]{\tan (x)}+\cos \left(\frac{\pi }{8}\right)\right)\right)-\\\sin \left(\frac{\pi }{8}\right) \tan ^{-1}\left(\sec \left(\frac{\pi }{8}\right) \left(\sqrt[4]{\tan (x)}-\sin \left(\frac{\pi }{8}\right)\right)\right)-\\\sin \left(\frac{\pi }{8}\right) \tan ^{-1}\left(\sec \left(\frac{\pi }{8}\right) \left(\sqrt[4]{\tan (x)}+\sin \left(\frac{\pi }{8}\right)\right)\right)+\\\frac{1}{2} \cos \left(\frac{\pi }{8}\right) \log \left(\sqrt{\tan (x)}+2 \sin \left(\frac{\pi }{8}\right) \sqrt[4]{\tan (x)}+1\right)-\\\frac{1}{2} \cos \left(\frac{\pi }{8}\right) \log \left(\sqrt{\tan (x)}-2 \sin \left(\frac{\pi }{8}\right) \sqrt[4]{\tan (x)}+1\right)-\\\frac{1}{2} \sin \left(\frac{\pi }{8}\right) \log \left(\sqrt{\tan (x)}+2 \cos \left(\frac{\pi }{8}\right) \sqrt[4]{\tan (x)}+1\right)+\\\frac{1}{2} \sin \left(\frac{\pi }{8}\right) \log \left(\sqrt{\tan (x)}-2 \cos \left(\frac{\pi }{8}\right) \sqrt[4]{\tan (x)}+1\right)+C$
I wouldn’t want to have to have worked it out by hand, but it doesn’t look impossible.
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