Integrand size = 17, antiderivative size = 85 \[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\frac {\cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} \csc ^{-1+n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]
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Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2667, 2657} \[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\frac {\cos ^{m-1}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} \csc ^{n-1}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]
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Rule 2657
Rule 2667
Rubi steps \begin{align*} \text {integral}& = \left (\csc ^n(e+f x) \sin ^n(e+f x)\right ) \int \cos ^m(e+f x) \sin ^{-n}(e+f x) \, dx \\ & = \frac {\cos ^{-1+m}(e+f x) \cos ^2(e+f x)^{\frac {1-m}{2}} \csc ^{-1+n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 2.39 (sec) , antiderivative size = 312, normalized size of antiderivative = 3.67 \[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=-\frac {2 (-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \cos ^m(e+f x) \csc ^n(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{f (-1+n) \left ((-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (m \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},1-m,1+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(1+m-n) \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},-m,2+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \left (\cos ^{m}\left (f x +e \right )\right ) \left (\csc ^{n}\left (f x +e \right )\right )d x\]
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\[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\int { \cos \left (f x + e\right )^{m} \csc \left (f x + e\right )^{n} \,d x } \]
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\[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\int \cos ^{m}{\left (e + f x \right )} \csc ^{n}{\left (e + f x \right )}\, dx \]
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\[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\int { \cos \left (f x + e\right )^{m} \csc \left (f x + e\right )^{n} \,d x } \]
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\[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\int { \cos \left (f x + e\right )^{m} \csc \left (f x + e\right )^{n} \,d x } \]
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Timed out. \[ \int \cos ^m(e+f x) \csc ^n(e+f x) \, dx=\int {\cos \left (e+f\,x\right )}^m\,{\left (\frac {1}{\sin \left (e+f\,x\right )}\right )}^n \,d x \]
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