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
