Table of Integrals, Series, and Products, 8th Edition

2.55-2.56 Rational functions of the sine and cosine

2.551

1. 

$\begin{array}{ccc}{\displaystyle \int \frac{A+B\sin x}{{(a+b\sin x)}^n}\text{d}x}\hfill & =\hfill & \frac{1}{(n-1)(a^2-b^2)}[\frac{(Ab-aB)\cos x}{{(a+b\sin x)}^{n-1}}\hfill \\ +{\displaystyle \int \frac{(Aa-Bb)(n-1)+(aB-bA)(n-2)\sin x}{{(a+b\sin x)}^{n-1}}\text{d}x}]\hfill \end{array}$

TI (358)a$\text{For }n=1:$

2. 

${\displaystyle \int \frac{A+B\sin x}{{(a+b\sin x)}^n}\text{d}x}=\frac{B}{b}x+\frac{Ab-aB}{b}{\displaystyle \int \frac{\text{d}x}{a+b\sin x}}$

(see 2.551 3)TI (342)

3. 

$\begin{array}{cccc}{\displaystyle \int \frac{\text{d}x}{a+b\sin x}}\hfill & =\hfill & \frac{2}{\sqrt{a^2-b^2}}\arctan \frac{a\tan \frac{x}{2}+b}{\sqrt{a^2-b^2}}\hfill & \left[a^2>b^2\right]\hfill \\ =\hfill & \frac{1}{\sqrt{b^2-a^2}}\ln \frac{a\tan \frac{x}{2}+b-\sqrt{b^2-a^2}}{a\tan \frac{x}{2}+b+\sqrt{b^2-a^2}}\hfill & \left[a^2<b^2\right]\hfill \end{array}$

2.552

1. 

$\begin{array}{l}{\displaystyle \int \frac{A+B\cos x}{{(a+b\sin x)}^n}\text{d}x}=-\frac{B}{(n-1)b{(a+b\sin x)}^{n-1}}+A{\displaystyle \int \frac{\text{d}x}{{(a+b\sin x)}^n}}\hfill \end{array}$

$\text{For }n=1:$(see 2.552