inverse trigonometric indefinite integral

Calculus Level 2

$\int\frac{\arctan(\sinh(x))}{\arcsin(\tanh(x))}\,dx = ?$

arcsinh(x) cos(x)+C x+C arccos(tanhx)+C

1 solution

Lorenzo Piras
Sep 20, 2018

$\arcsin t=\arctan \frac t{\sqrt{1-t^2}}$

then

$\arcsin\tanh x=\arctan\frac{\tanh x}{\sqrt{1-\tanh^2x}}=\arctan\frac{\sinh x}{\sqrt{\cosh^2x-\sinh^2x}}=\arctan\sinh x.$ so the rusult is just $x+C$