Primitive of Cube of Tangent of a x/Proof 2

Theorem

$\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$

$\ds I_n$

$=$

$\ds \int \map {\tan^n} {a x} \rd x$

$\ds $

$=$

$\ds \frac {\map {\tan^{n - 1} } {a x} } {a \paren {n - 1} } - I_{n - 2}$

$\ds I_1$

$=$

$\ds -\frac 1 a \ln \size {\map \cos {a x} } + C$

$\ds \leadsto \ \ $

$\ds I_3$

$=$

$\ds \frac {\map {\tan^2} {a x} } {2 a} + \frac 1 a \ln \size {\map \cos {a x} } + C'$

$\blacksquare$