What is the integral of 1/(sinh^2(x)cosh^2(x)) ?

The integral of 1/(sinh^2(x)cosh^2(x)) is -sech^2(x)coth(x)-2tanh(x)+C

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integral of 1/(sinh^2(x)cosh^2(x))

\int \frac{1}{sinh ^{2} ( x ) cosh ^{2} ( x )} d x

- sech^{2} (x) coth (x) - 2 tanh (x) + C

Solution steps

\int \frac{1}{sinh ^{2} ( x ) cosh ^{2} ( x )} d x

Rewrite using trig identities

= \int \frac{sech ^{2} ( x )}{sinh ^{2} ( x )} d x

Use the Hyperbolic identity:

\frac{1}{sinh ( x )} = csch (x)

= \int sech^{2} (x) csch^{2} (x) d x

Apply Integration By Parts

= - sech^{2} (x) coth (x) - \int 2 sech^{2} (x) tanh (x) coth (x) d x

\int 2 sech^{2} (x) tanh (x) coth (x) d x = 2 tanh (x)

= - sech^{2} (x) coth (x) - 2 tanh (x)

Add a constant to the solution

= - sech^{2} (x) coth (x) - 2 tanh (x) + C

What is the integral of 1/(sinh^2(x)cosh^2(x)) ?
The integral of 1/(sinh^2(x)cosh^2(x)) is -sech^2(x)coth(x)-2tanh(x)+C