Integrand size = 19, antiderivative size = 79 \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=-\frac {(d \cot (e+f x))^{1+n} \csc ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {4+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {4+n}{2}}}{d f (1+n)} \]
[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} \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=-\frac {\csc ^3(e+f x) \sin ^2(e+f x)^{\frac {n+4}{2}} (d \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {n+4}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]
[In]
[Out]
Rule 2697
Rubi steps \begin{align*} \text {integral}& = -\frac {(d \cot (e+f x))^{1+n} \csc ^3(e+f x) \operatorname {Hypergeometric2F1}\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*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 12.60 (sec) , antiderivative size = 784, normalized size of antiderivative = 9.92 \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=-\frac {\cot ^2\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (-1-\frac {n}{2},-n,-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n}}{f (8+4 n)}+\frac {8 (-4+n) \cos ^6\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \csc ^2(e+f x) \left (n \operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-(-2+n) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-n,-\frac {n}{2},1-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{f (-2+n) n \left (-8 n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,1,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )-8 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,2,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )+(-4+n) \left (4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^n-\operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2(e+f x)\right )\right )}+\frac {(d \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-n,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n} \tan ^2\left (\frac {1}{2} (e+f x)\right )}{f (8-4 n)}+\frac {(-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (d \cot (e+f x))^n \sin ^2\left (\frac {1}{2} (e+f x)\right )}{f (4-2 n) \left ((-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 \left (n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,1,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\operatorname {AppellF1}\left (2-\frac {n}{2},-n,2,3-\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 ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
[In]
[Out]
\[\int \left (d \cot \left (f x +e \right )\right )^{n} \csc \left (f x +e \right )^{3}d x\]
[In]
[Out]
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{n} \csc ^{3}{\left (e + f x \right )}\, dx \]
[In]
[Out]
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int \frac {{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^3} \,d x \]
[In]
[Out]