Evaluate `int (log_(e)(1+sin^(2)x))/(cos^(2)x) dx`

Explanation

`I=int log_(e) (1+sin^(2)x).sec^(2)x dx`
`= log_(e) (1+sin^(2)x). tanx -int(2sinx cos x)/(1+sin^(2)x)tanx dx`
`=tanx xx log_(e) (1+sin^(2)x)-2int(sin^(2)x)/(1+sin^(2)x)dx`
`=tanx xx log_(e) (1+sin^(2)x)-2int(1-(1)/(1+sin^(2)x))dx`
`=tanx xx log_(e) (1+sin^(2)x)-2x+2 int (sec^(2) x dx)/(sec^(2)x +tan^(2)x)`
`=tanx xx log_(e) (1+sin^(2)x)-2x +2int(dt)/(1+2t^(2)) " " ("where "t=tanx)`
`=tanx xx log_(e) (1+sin^(2)x)-2x+2.(1)/(sqrt(2))"tan"^(-1)sqrt(2)t +c`
`=tanx xx log_(e) (1+sin^(2)x)-2x+sqrt(2)tan^(-1)(sqrt(2)tanx)+c`