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

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## Theorem

- $\displaystyle \int \frac {\mathrm d x} {p \sin a x + q \cos a x + \sqrt {p^2 + q^2} } = \frac {-1} {a \sqrt {p^2 + q^2} } \tan \left({\frac \pi 4 - \frac {a x + \arctan \frac q p} 2}\right) + C$

## Proof

## Sources

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