Integrand size = 19, antiderivative size = 78 \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=-\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {6-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n}{f (1-n)} \]
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Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2681, 2656} \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=-\frac {\cos (e+f x) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {6-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n)} \]
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Rule 2656
Rule 2681
Rubi steps \begin{align*} \text {integral}& = \left (\cos ^n(e+f x) \sin ^{-n}(e+f x) (b \tan (e+f x))^n\right ) \int \cos ^{-n}(e+f x) \sin ^{-5+n}(e+f x) \, dx \\ & = -\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {6-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{-n/2} (b \tan (e+f x))^n}{f (1-n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 14.17 (sec) , antiderivative size = 1017, normalized size of antiderivative = 13.04 \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\frac {3 \cot ^2\left (\frac {1}{2} (e+f x)\right ) \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 (b \tan (e+f x))^n}{16 f (-2+n)}+\frac {\cot ^2\left (\frac {1}{2} (e+f x)\right ) \left ((-2+n) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-2+\frac {n}{2},n,-1+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(-4+n) \operatorname {Hypergeometric2F1}\left (-1+\frac {n}{2},n,\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^n (b \tan (e+f x))^n}{16 f (-4+n) (-2+n)}+\frac {3 (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 ) \sin ^2(e+f x) (b \tan (e+f x))^n}{16 f (2+n) \left (2 \left (\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 )-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 )\right ) (-1+\cos (e+f x))+(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 ) (1+\cos (e+f x))\right )}+\frac {3 \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 ) (b \tan (e+f x))^n}{16 f (2+n)}+\frac {\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 ) \left ((4+n) \operatorname {Hypergeometric2F1}\left (1+\frac {n}{2},n,2+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(2+n) \operatorname {Hypergeometric2F1}\left (2+\frac {n}{2},n,3+\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (b \tan (e+f x))^n}{16 f (2+n) (4+n)}+\frac {3 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \cot \left (\frac {1}{2} (e+f x)\right ) \left ((2+n) \operatorname {Hypergeometric2F1}\left (\frac {n}{2},n,1+\frac {n}{2},\tan ^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 ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (b \tan (e+f x))^n}{f n (2+n) \left (-8 \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 ) \tan \left (\frac {1}{2} (e+f x)\right )+\frac {8 \left (2 \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 )-2 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 )+(4+n) \cot ^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}\right ) \tan ^3\left (\frac {1}{2} (e+f x)\right )}{4+n}\right )} \]
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\[\int \left (\csc ^{5}\left (f x +e \right )\right ) \left (b \tan \left (f x +e \right )\right )^{n}d x\]
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\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5} \,d x } \]
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\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int \left (b \tan {\left (e + f x \right )}\right )^{n} \csc ^{5}{\left (e + f x \right )}\, dx \]
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\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5} \,d x } \]
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\[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int { \left (b \tan \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{5} \,d x } \]
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Timed out. \[ \int \csc ^5(e+f x) (b \tan (e+f x))^n \, dx=\int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^5} \,d x \]
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