Primitive of Reciprocal of p by Sine of a x plus q by Cosine of a x plus or minus Root of p squared plus q squared

Theorem

$\ds \int \frac {\d x} {p \sin a x + q \cos a x \pm \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \map \tan {\frac \pi 4 \mp \frac {a x + \arctan \frac q p} 2} + C$

Primitive of $\dfrac 1 {p \sin a x + q \cos a x + \sqrt {p^2 + q^2}}$

$\ds \int \frac {\d x} {p \sin a x + q \cos a x + \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \map \tan {\frac \pi 4 - \frac {a x + \arctan \frac q p} 2} + C$

Primitive of $\dfrac 1 {p \sin a x + q \cos a x - \sqrt {p^2 + q^2}}$

$\ds \int \frac {\d x} {p \sin a x + q \cos a x - \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \map \tan {\frac \pi 4 + \frac {a x + \arctan \frac q p} 2} + C$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.422$