The easiest way to integrate (or differentiate) the hyperbolic functions is to use their definitions:
#sinh(x)=(e^x-e^(-x))/2#
#cosh(x)=(e^x+e^(-x))/2#
#tanh(x)=sinh(x)/cosh(x)=(e^x-e^(-x))/(e^x+e^(-x))#
#coth(x)=cosh(x)/sinh(x)=(e^x+e^(-x))/(e^x-e^(-x))#
From here, it should be reasonably straightforward to show that
#int sinh(x)dx = cosh(x) + C#
#int cosh(x)dx = sinh(x) + C#
#int tanh(x)dx = ln(cosh((x)) + C#
#int coth(x)dx = ln(sinh(x))+ C#
where C is the constant of integration. I will show the first two here:
#int sinh(x)dx = int (e^x-e^-x)/2 = int e^x/2-e^(-x)/2dx#
#=e^x/2-(-e^-x)/2 + C# (where #C# is the constant of integration)
#=e^x/2+e^(-x)/2 + C#
#=cosh(x)+C#.
Similarly,
#int cosh(x)dx = int e^x/2+e^(-x)/2dx#
#=e^x/2+(-e^-x)/2 + C#
#=e^x/2-e^-x/2 + C#
#=sinh(x) + C#.
