Integrand size = 17, antiderivative size = 73 \[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=-\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin (e+f x) \sin ^2(e+f x)^{n/2}}{d f (1+n)} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2697} \[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=-\frac {\sin (e+f x) \sin ^2(e+f x)^{n/2} (d \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = -\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin (e+f x) \sin ^2(e+f x)^{n/2}}{d f (1+n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 1.43 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.62 \[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=-\frac {8 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,2,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \sin ^2\left (\frac {1}{2} (e+f x)\right )}{f (-2+n) \left (2 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,2,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 \left (n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,2,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,3,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\cos (e+f x))\right )} \]
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\[\int \left (d \cot \left (f x +e \right )\right )^{n} \sin \left (f x +e \right )d x\]
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\[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right ) \,d x } \]
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\[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{n} \sin {\left (e + f x \right )}\, dx \]
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\[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right ) \,d x } \]
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\[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right ) \,d x } \]
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Timed out. \[ \int (d \cot (e+f x))^n \sin (e+f x) \, dx=\int \sin \left (e+f\,x\right )\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]
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