Integral of $$$\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)}$$$

Find $$$\int \tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)}\, dx$$$.

Simplify the integrand:

$${\color{red}{\int{\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)} d x}}} = {\color{red}{\int{1 d x}}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$${\color{red}{\int{1 d x}}} = {\color{red}{x}}$$

Therefore,

$$\int{\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)} d x} = x$$

Add the constant of integration:

$$\int{\tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)} d x} = x+C$$

$$$\int \tan^{3}{\left(x \right)} \cot^{3}{\left(x \right)}\, dx = x + C$$$A