What is the integral of (cot^2(x)+1)^2csc^2(x) ?

The integral of (cot^2(x)+1)^2csc^2(x) is -(cos(x)csc^5(x))/5+4/5 (-1/3 csc^3(x)cos(x)-2/3 cot(x))+C

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integral of (cot^2(x)+1)^2csc^2(x)

\int (cot^{2} (x) + 1)^{2} csc^{2} (x) d x

- \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x)) + C

Solution steps

\int (cot^{2} (x) + 1)^{2} csc^{2} (x) d x

Rewrite using trig identities

= \int csc^{6} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} \cdot \int csc^{4} (x) d x

\int csc^{4} (x) d x = - \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x)

= - \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x))

Add a constant to the solution

= - \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x)) + C

\int x (2 x + 3)^{7} d x

\int a x^{7} d x

\int \frac{cot ( t ) csc ( t )}{csc ^{7} ( t )} d t

\int - tan (2 x + 1) ln (cos (2 x + 1)) d x

\int \frac{2 x - 7}{x ^{2} - 12 x + 32} d x

What is the integral of (cot^2(x)+1)^2csc^2(x) ?
The integral of (cot^2(x)+1)^2csc^2(x) is -(cos(x)csc^5(x))/5+4/5 (-1/3 csc^3(x)cos(x)-2/3 cot(x))+C