Evaluate: `intx^3tan^(-1)x\ dx`

Explanation

Let `I=int x^{3} tan ^{-1} x d x`

Using Integration by parts,


`frac{d}{d x} tan ^{-1} x=frac{1}{2(1+x^{2})} `
`
=tan ^{-1} x int x^{3} d x-int(frac{1}{1+x^{2}}) int x^{3} d x`
`
=tan ^{-1} x frac{x^{4}}{4}-frac{1}{4} int frac{x^{4}}{1+x^{2}} d x `
`
frac{1}{4} int frac{x^{4}}{1+x^{2}} d x=frac{1}{4}[int frac{1}{1+x^{2}} d x+(x^{2}-1) d x] `
`
=frac{1}{4}[tan ^{-1} x+frac{x^{3}}{3}-x] `
`
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