Optimal. Leaf size=25 \[ -\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.042452, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 32} \[ -\frac{(d \cot (e+f x))^{n+1}}{d f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2607
Rule 32
Rubi steps
\begin{align*} \int (d \cot (e+f x))^n \csc ^2(e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int (-d x)^n \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac{(d \cot (e+f x))^{1+n}}{d f (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0193327, size = 26, normalized size = 1.04 \[ -\frac{\cot (e+f x) (d \cot (e+f x))^n}{f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 26, normalized size = 1. \begin{align*} -{\frac{ \left ( d\cot \left ( fx+e \right ) \right ) ^{1+n}}{fd \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.72804, size = 97, normalized size = 3.88 \begin{align*} -\frac{\left (\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{{\left (f n + f\right )} \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot{\left (e + f x \right )}\right )^{n} \csc ^{2}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]