Evaluate: `inttan^5xdx`

Explanation

We have given`inttan^5xdx`

Let
`I=inttan^5xdx=inttan^2xtan^3xdx=inttan^3x(sec^2x−1)dx`

`I=int(tan^3xsec^2x−tan^3x)dx`

`I=int(tan^3xsec^2x−tanx(sec^2x−1))dx`

`I=int(tan^3xsec^2x−tanxsec^2x+tanx)dx`

`I=(tan^4x)/4-(tan^2x)/2+ln∣secx∣+C`