What is the integral of (csch(3tan(x)))/(cos^2(x)) ?

The integral of (csch(3tan(x)))/(cos^2(x)) is 1/3 ln(tanh((3tan(x))/2))+C

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integral of (csch(3tan(x)))/(cos^2(x))

\int \frac{csch ( 3 tan ( x ) )}{cos ^{2} ( x )} d x

\frac{1}{3} ln (tanh (\frac{3 tan ( x )}{2})) + C

Solution steps

\int \frac{csch ( 3 tan ( x ) )}{cos ^{2} ( x )} d x

Apply u-substitution

: \int csch (3 u) d u

= \int csch (3 u) d u

Apply u-substitution

: \int csch (v) \frac{1}{3} d v

= \int csch (v) \frac{1}{3} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int csch (v) d v

Use the common integral:

\int csch (v) d v = ln (tanh (\frac{v}{2}))

= \frac{1}{3} ln (tanh (\frac{v}{2}))

Substitute back

= \frac{1}{3} ln (tanh (\frac{3 tan ( x )}{2}))

Add a constant to the solution

= \frac{1}{3} ln (tanh (\frac{3 tan ( x )}{2})) + C

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What is the integral of (csch(3tan(x)))/(cos^2(x)) ?
The integral of (csch(3tan(x)))/(cos^2(x)) is 1/3 ln(tanh((3tan(x))/2))+C