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

GU (351)(18)

2.452

1. 

$\begin{array}{l}{\displaystyle \int \frac{A+B\text{cosh}x+C\text{sinh}x}{\left(a_1+b_1\text{cosh}x+c_1\text{sinh}x\right)\left(a_2+b_2\text{cosh}x+c_2\text{sinh}x\right)}\text{d}x}\hfill \\ =A_{0}ln\frac{\left(a_1+b_1\text{cosh}x+c_1\text{sinh}x\right)}{\left(a_2+b_2\text{cosh}x+c_2\text{sinh}x\right)}+A_1{\displaystyle \int \frac{\text{d}x}{\left(a_1+b_1\text{cosh}x+c_1\text{sinh}x\right)}+A_2{\displaystyle \int \frac{\text{d}x}{\left(a_2+b_2\text{cosh}x+c_2\text{sinh}x\right)}}}\hfill \end{array}$

GU (351)(19)where$\begin{array}{cccccc}{A}_0\hfill & =\hfill & \frac{\left|\begin{array}{ccc}{a}_1\hfill & b_1\hfill & c_1\hfill \\ A\hfill & B\hfill & C\hfill \\ a_2\hfill & b_2\hfill & c_2\hfill \end{array}\right|}{{\left|\begin{array}{cc}{a}_1\hfill & b_1\hfill \\ a_2\hfill & b_2\hfill \end{array}\right|}^2+{\left|\begin{array}{cc}{b}_1\hfill & c_1\hfill \\ b_2\hfill & c_2\hfill \end{array}\right|}^2-{\left|\begin{array}{cc}{c}_1\hfill & a_1\hfill \\ c_2\hfill & a_2\hfill \end{array}\right|}^2}\hfill & A_1\hfill & =\hfill & \frac{\left|\begin{array}{ccc}{a}_1\hfill & b_1\hfill & c_1\hfill \\ \left|\begin{array}{cc}{b}_1\hfill & c_1\hfill \\ B\hfill & C\hfill \end{array}\right|\hfill & \left|\begin{array}{cc}{c}_1\hfill & a_1\hfill \\ C\hfill & A\hfill \end{array}\right|\hfill & \left|\begin{array}{cc}{a}_1\hfill & b_1\hfill \\ A\hfill & B\hfill \end{array}\right|\hfill \\ a_2\hfill & b_2\hfill & c_2\hfill \end{array}\right|}{{\left|\begin{array}{cc}{a}_1\hfill & b_1\hfill \\ a_2\hfill & b_2\hfill \end{array}\right|}^2+{\left|\begin{array}{cc}{b}_1\hfill & c_1\hfill \\ b_2\hfill & c_2\hfill \end{array}\right|}^2-{\left|\begin{array}{cc}{c}_1\hfill & a_1\hfill \\ c_2\hfill & a_2\hfill \end{array}\right|}^2},\hfill \\ A_2\hfill & =\hfill & \frac{\left|\begin{array}{ccc}{a}_1\hfill & b_1\hfill & c_1\hfill \\ \left|\begin{array}{cc}C\hfill & B\hfill \\ c_2\hfill & b_2\hfill \end{array}\right|\hfill & \left|\begin{array}{cc}C\hfill & A\hfill \\ c_2\hfill & a_2\hfill \end{array}\right|\hfill & \left|\begin{array}{cc}B\hfill & A\hfill \\ b_2\hfill & a_2\hfill \end{array}\right|\hfill \\ a_2\hfill & b_2\hfill & c_2\hfill \end{array}\right|}{{\left|\begin{array}{cc}{a}_1\hfill & b_1\hfill \\ a_2\hfill & b_2\hfill \end{array}\right|}^2+{\left|\begin{array}{cc}{b}_1\hfill & c_1\hfill \\ b_2\hfill & c_2\hfill \end{array}\right|}^2-{\left|\begin{array}{cc}{c}_1\hfill & a_1\hfill \\ c_2\hfill & a_2\hfill \end{array}\right|}^2}\hfill & \left[{\left|\begin{array}{cc}{a}_1\hfill & b_1\hfill \\ a_2\hfill & b_2\hfill \end{array}\right|}^2+{\left|\begin{array}{cc}{b}_1\hfill & c_1\hfill \\ b_2\hfill & c_2\hfill \end{array}\right|}^2\ne {\left|\begin{array}{cc}{c}_1\hfill & a_1\hfill \\ c_2\hfill & a_2\hfill \end{array}\right|}^2\right].\hfill \end{array}$

2. 

$\begin{array}{c}{\displaystyle \int \frac{A}{}}\hfill \end{array}$