Интегралы от гиперболических функций

Неопределённые интегралы от гиперболических функций.

\int ch a x d x = \frac{1}{a} sh a x + C

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

\int sh a x d x = \frac{1}{a} ch a x + C

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

\int th a x d x = \frac{1}{a} ln ch a x + C

\int e^{a x} th 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 arct g [ e ^{a x} ]}{a} a = b + C

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

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

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

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

\int sh a x ch a x d x = \frac{1}{4 a} ch 2 a x + C

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