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\int \tan ^{3}x\sec ^{3}x\,dx\,

With an odd number of \tan x, greater than 1, turn all but 1 into \sec ^{2}x and we'll have a whole bunch of \sec x and the derivative, \sec x\tan x.

\int \tan ^{3}x\sec ^{3}x\,dx=\int \sec ^{2}x(\sec ^{2}x-1)\sec x\tan x\,dx=\int (\sec ^{4}x-\sec ^{2}x)\sec x\tan x\,dx\,

Again, we can write out the steps of a substitution but hopefully at some point you can just see them without writing them out. Thus

\int \tan ^{3}x\sec ^{3}x\,dx={\frac  {1}{5}}\sec ^{5}x-{\frac  {1}{3}}\sec ^{3}x+C\,

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