Integrals of Hyperbolic Functions

Indefinite integrals of the hyperbolic functions.

\int cosh a x d x = \frac{1}{a} sinh a x + C

\int e^{a x} cosh b x d x = ⎩ ⎨ ⎧ \frac{e ^{a x}}{a ^{2} - b ^{2}} [a cosh b x - b sinh b x] a \neq = b \frac{e ^{2 a x}}{4 a} + \frac{x}{2} a = b + C

\int sinh a x d x = \frac{1}{a} cosh a x + C

\int e^{a x} sinh b x d x = ⎩ ⎨ ⎧ \frac{e ^{a x}}{a ^{2} - b ^{2}} [- b cosh b x + a sinh b x] a \neq = b \frac{e ^{2 a x}}{4 a} - \frac{x}{2} a = b + C

\int tanh a x d x = \frac{1}{a} ln cosh a x + C

\int e^{a x} tanh b x d x = ⎩ ⎨ ⎧ \frac{e ^{(a + 2 b) x}}{( a + 2 b )}_{2} F_{1} [1 + \frac{a}{2 b}, 1, 2 + \frac{a}{2 b}, - e^{2 b x}] - \frac{1}{a} e^{a x}_{2} F_{1} [1, \frac{a}{2 b}, 1 + \frac{a}{2 b}, - e^{2 b x}] a \neq = b \frac{e ^{a x} - 2 arctan [ e ^{a x} ]}{a} a = b + C

\int cos a x cosh b x d x = \frac{1}{a ^{2} + b ^{2}} [a sin a x cosh b x + b cos a x sinh b x] + C

\int cos a x sinh b x d x = \frac{1}{a ^{2} + b ^{2}} [b cos a x cosh b x + a sin a x sinh b x] + C

\int sin a x cosh b x d x = \frac{1}{a ^{2} + b ^{2}} [- a cos a x cosh b x + b sin a x sinh b x] + C

\int sin a x sinh b x d x = \frac{1}{a ^{2} + b ^{2}} [b cosh b x sin a x - a cos a x sinh b x] + C

\int sinh a x cosh a x d x = \frac{1}{4 a} cosh 2 a x + C

\int sinh a x cosh b x d x = \frac{1}{a ^{2} - b ^{2}} [a cosh a x cosh b x - b sinh a x sinh b x] + C, a \neq = b