Optimal. Leaf size=79 \[ -\frac{\csc ^3(e+f x) \sin ^2(e+f x)^{\frac{n+4}{2}} (d \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{n+4}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0405785, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2617} \[ -\frac{\csc ^3(e+f x) \sin ^2(e+f x)^{\frac{n+4}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{n+4}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2617
Rubi steps
\begin{align*} \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx &=-\frac{(d \cot (e+f x))^{1+n} \csc ^3(e+f x) \, _2F_1\left (\frac{1+n}{2},\frac{4+n}{2};\frac{3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac{4+n}{2}}}{d f (1+n)}\\ \end{align*}
Mathematica [B] time = 6.51261, size = 190, normalized size = 2.41 \[ -\frac{\tan ^2\left (\frac{1}{2} (e+f x)\right ) (d \cot (e+f x))^n \left (\cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )\right )^{-n} \left ((n-2) n \cot ^4\left (\frac{1}{2} (e+f x)\right ) \text{Hypergeometric2F1}\left (-\frac{n}{2}-1,-n,-\frac{n}{2},\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+(n+2) \left (n \text{Hypergeometric2F1}\left (1-\frac{n}{2},-n,2-\frac{n}{2},\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+2 (n-2) \cot ^2\left (\frac{1}{2} (e+f x)\right ) \text{Hypergeometric2F1}\left (-n,-\frac{n}{2},1-\frac{n}{2},\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )}{4 f n \left (n^2-4\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.551, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( \csc \left ( fx+e \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3}, x\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 ^{3}{\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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]