Given: #intsech^6(x)dx#
Use the hyperbolic secant reduction formula,
#intsech^m(x)dx = (sech(x)sech^(m-1)(x))/(m-1)+ (m-2)/(m-1)intsech^(m-2)(x)dx#
, where m = 6:
#intsech^6(x)dx=(sech(x)sech^(5)(x))/(5)+ 4/5intsech^4(x)dx#
Use the hyperbolic secant reduction formula,
#intsech^m(x)dx = (sech(x)sech^(m-1)(x))/(m-1)+ (m-2)/(m-1)intsech^(m-2)(x)dx#
, where m = 4:
#intsech^6(x)dx=(sech(x)sech^(5)(x))/(5)+ 4/5{(sech(x)sech^(3)(x))/(3)+ 2/3intsech^2(x)dx}#
The last integral is trivial:
#intsech^6(x)dx=(sech(x)sech^(5)(x))/(5)+ 4/5{(sech(x)sech^(3)(x))/(3)+ 2/3tanh(x)+ C}#
Distribute the #4/5#
#intsech^6(x)dx=(sech(x)sech^(5)(x))/(5)+ (4sech(x)sech^(3)(x))/15+ 8/15tanh(x)+ C#
