Integrand size = 19, antiderivative size = 79 \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+n),\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^3(e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-2+n)}}{d f (1+n)} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697} \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {\sin ^3(e+f x) \sin ^2(e+f x)^{\frac {n-2}{2}} (d \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n-2}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]
[In]
[Out]
Rule 2697
Rubi steps \begin{align*} \text {integral}& = -\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+n),\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^3(e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-2+n)}}{d f (1+n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 3.10 (sec) , antiderivative size = 477, normalized size of antiderivative = 6.04 \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {4 (-4+n) \left (\operatorname {AppellF1}\left (1-\frac {n}{2},-n,3,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-\operatorname {AppellF1}\left (1-\frac {n}{2},-n,4,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x)}{f (-2+n) \left (2 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,3,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,4,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 \left (n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,3,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,4,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+3 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,4,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,5,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\cos (e+f x))\right )} \]
[In]
[Out]
\[\int \left (d \cot \left (f x +e \right )\right )^{n} \sin \left (f x +e \right )^{3}d x\]
[In]
[Out]
\[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
\[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^3\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]
[In]
[Out]