INTEGRALS CONTAINING RECIPROCALS OF HYPERBOLIC FUNCTIONS

1.

$\displaystyle\int\sinh^{-1}\displaystyle \frac{x}{a}dx=x\sinh^{-1}\displaystyle \frac{x}{a}-\displaystyle \sqrt{x^2+a^2}$

2.

$\displaystyle\int x\sinh^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \f... ...\displaystyle \frac{x}{a}-\displaystyle \frac{x\displaystyle \sqrt{x^2+a^2}}{4}$

3.

$\displaystyle\int x^2\sinh^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{... ...tyle \frac{x}{a}+\displaystyle \frac{(2a^2-x^2)\displaystyle \sqrt{x^2+a^2}}{9}$

4.

$\displaystyle\int\displaystyle \frac{\sinh^{-1}(x/a)}{x}dx=\left\{ \begin{array... ...dot 4\cdot 6\cdot 6\cdot 6}- \cdot\cdot\cdot&\ast \ast \ast \end{array}\right.$

where $\ast$ is |x| < a, $\ast \ast$ is x>a, and $\ast \ast \ast$ is x<-a.

5.

$\displaystyle\int\displaystyle \frac{\sinh^{-1}(x/a)}{x^2}dx=-\displaystyle \fr... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^2+a^2}}{x}\right)$

6.

$\displaystyle\int\cosh^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{ll... ...isplaystyle \sqrt{x^2-a^2},&\hspace{.2in} \cosh^{-1}(x/a)<0 \end{array}\right.$

7.

$\displaystyle\int x\cosh^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{... ...displaystyle \sqrt{x^2-a^2},&\hspace{.2in}\cosh^{-1}(x/a)<0 \end{array}\right.$

8.

$\displaystyle\int x^2\cosh^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array... ...9}(x^2+2a^2)\displaystyle \sqrt{x^2-a^2},&\cosh^{-1}(x/a)<0 \end{array}\right.$

9.

$\displaystyle\int\displaystyle \frac{\cosh^{-1}(x/a)}{x}dx=\pm\left[\displaysty... ...{1\cdot 3\cdot 5(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6}+\cdot\cdot\cdot \right]$

10.

$\displaystyle\int\displaystyle \frac{\cosh^{-1}(x/a)}{x^2}dx=-\displaystyle \fr... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^2+a^2}}{x}\right)$

11.

$\displaystyle\int\tanh^{-1}\displaystyle \frac{x}{a}dx=x\tanh^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{a}{2}\ln(a^2-x^2)$

12.

$\displaystyle\int x\tanh^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{ax}{2}+\displaystyle \frac{1}{2}(x^2-a^2)\tanh^{-1}\displaystyle \frac{x}{a}$

13.

$\displaystyle\int x^2\tanh^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{... ...}{3}\tanh^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{a^3}{6}\ln(a^2-x^2)$

14.

$\displaystyle\int\displaystyle \frac{\tanh^{-1}(x/a)}{x}dx=\displaystyle \frac{... ...playstyle \frac{(x/a)^3}{3^2}+\displaystyle \frac{(x/a)^5}{5^2}+\cdot\cdot\cdot$

15.

$\displaystyle\int\displaystyle \frac{\tanh^{-1}(x/a)}{x^2}dx=-\displaystyle \fr... ...{x}+\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2}{a^2-x^2}\right)$

16.

$\displaystyle\int\coth^{-1}\displaystyle \frac{x}{a}dx=x\coth^{-1}x+\displaystyle \frac{a}{2}\ln(x^2-a^2)$

17.

$\displaystyle\int x\coth^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{ax}{2}+\displaystyle \frac{1}{2}(x^2-a^2)\coth^{-1}\displaystyle \frac{x}{a}$

18.

$\displaystyle\int x^2\coth^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{... ...}{3}\coth^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{a^3}{6}\ln(x^2-a^2)$

19.

$\displaystyle\int\displaystyle \frac{\coth^{-1}(x/a)}{x}dx=-\left(\displaystyle... ...le \frac{(a/x)^3}{3^2}+\displaystyle \frac{(a/x)^5}{5^2}+\cdot\cdot\cdot\right)$

20.

$\displaystyle\int\displaystyle \frac{\coth^{-1}(x/a)}{x^2}dx=-\displaystyle \fr... ...{x}+\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2}{x^2-a^2}\right)$

21.

$\displaystyle\int\cosh(x/a)dx=\left\{ \begin{array}{ll} x\cosh(x/a)+a\sin^{-1}(... ... \\ x\cosh(x/a)-a\sin^{-1}(x/a),&\hspace{.2in}\cosh(x/a)<0 \end{array}\right.$

22.

$\displaystyle\int x\cosh(x/a)dx=\left\{ \begin{array}{ll} \displaystyle \frac{x... ...}{2}\displaystyle \sqrt{a^2-x^2},&\hspace{.2in}\cosh(x/a)<0 \end{array}\right.$

23.

$\displaystyle\int\displaystyle \frac{\cosh(x/a)}{x}dx=\left\{ \begin{array}{ll}... ...4\cdot 4\cdot 4}+\cdot\cdot\cdot,&\hspace{.2in}\cosh(x/a)<0 \end{array}\right.$

24.

$\displaystyle\int\sinh(x/a)dx=x\sinh(x/a)\pm a\sinh^{-1}(x/a)$

25.

$\displaystyle\int x\sinh(x/a)dx=\displaystyle \frac{x^2}{2}\sinh(x/a)\pm\displaystyle \frac{a\displaystyle \sqrt{x^2+a^2}}{2}$

26.

$\displaystyle\int\displaystyle \frac{\sinh(x/a)}{x}dx=\left\{ \begin{array}{ll}... ...x)^5}{2\cdot 4\cdot 5\cdot 5}-\cdot\cdot\cdot& \mid x\mid>a \end{array}\right.$

27.

$\displaystyle\int x^m\sinh^{-1}(x/a)dx=\displaystyle \frac{x^{m+1}}{m+1}\sinh^{... ...e \frac{1}{m+1}\int\displaystyle \frac{x^{m+1}}{\displaystyle \sqrt{x^2+a^2}}dx$

28.

$\displaystyle\int x^m\cosh^{-1}(x/a)dx=\left\{ \begin{array}{ll} \displaystyle ... ...x^{m+1}}{\displaystyle \sqrt{x^2-a^2}}dx& \cosh^{-1}(x/a)<0 \end{array}\right.$

29.

$\displaystyle\int x^m\tanh^{-1}(x/a)dx=\displaystyle \frac{x^{m+1}}{m+1}\tanh^{-1}(x/a)-\displaystyle \frac{a}{m+1}\int\displaystyle \frac{x^{m+1}}{a^2-x^2}dx$

30.

$\displaystyle\int x^m\coth^{-1}(x/a)dx=\displaystyle \frac{x^{m+1}}{m+1}\coth^{-1}(x/a)-\displaystyle \frac{a}{m+1}\int\displaystyle \frac{x^{m+1}}{a^2-x^2}dx$

31.

$\displaystyle\int x^m\cosh(x/a)dx=\left\{ \begin{array}{ll} \displaystyle \frac... ...ystyle \sqrt{a^2-x^2}}& \mbox{\hspace*{0.5cm}} \cosh(x/a)<0 \end{array}\right.$

32.

$\displaystyle\int x^m\sinh(x/a)dx=\displaystyle \frac{x^{m+1}}{m+1}\sinh(x/a)\p... ...tyle \frac{a}{m+1}\int\displaystyle \frac{x^m dx}{\displaystyle \sqrt{x^2+a^2}}$