∫ 1/ (cos3x – cosx ) dx , please say how to simpl

Question

∫ 1/ (cos3x – cosx ) dx , please say how to simplyfy, thank you.

Arun · Accepted Answer

dx/cos3x-cosx =dx/4cos^3x-4cosx by using identity cos3x =4cos^3x - 3cosx

= dx/4.cosx(cos^2x-1)

=cosxdx/4cos^x(cos^x-1)

=-cosxdx/4sin^2x(1-sin^x)

now putting sinx=t,cosxdx=dt

=-dt/4(1-t^2)t^2

=1/4 {-dt/t^2 -dt/1-t^2 }

=1/4 .{1/t + log1-t/1+t } +c

=1/4 .(cosecx + log1-sinx/1+sinx } +c

∫ 1/ (cos3x – cosx ) dx , please say how to simplyfy, thank you.