Primitive of x over Cosine of a x

Theorem

$\ds \int \frac {x \rd x} {\cos a x}$

$=$

$\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C$

$\ds $

$=$

$\ds \dfrac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 + \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} + \cdots} + C$

where $E_n$ denotes the $n$th Euler number.

$\ds \int \frac {x \rd x}{\cos a x}$

$=$

$\ds \int x \sec a x \rd x$

$\ds $

$=$

$\ds \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_n \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + C$

$\blacksquare$