Integrand size = 21, antiderivative size = 91 \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=\frac {a b (a \cos (e+f x))^{-1+m} \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]
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Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 2657} \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=\frac {a b \cos ^2(e+f x)^{\frac {1-m}{2}} (a \cos (e+f x))^{m-1} (b \csc (e+f x))^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {1-m}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n)} \]
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Rule 2657
Rule 2667
Rubi steps \begin{align*} \text {integral}& = \left (b^2 (b \csc (e+f x))^{-1+n} (b \sin (e+f x))^{-1+n}\right ) \int (a \cos (e+f x))^m (b \sin (e+f x))^{-n} \, dx \\ & = \frac {a b (a \cos (e+f x))^{-1+m} \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{-1+n} \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 = 1.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.47 \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=-\frac {2 (-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (a \cos (e+f x))^m (b \csc (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right )}{f (-1+n) \left ((-3+n) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {n}{2},-m,1+m-n,\frac {3}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (m \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},1-m,1+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(1+m-n) \operatorname {AppellF1}\left (\frac {3}{2}-\frac {n}{2},-m,2+m-n,\frac {5}{2}-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \left (\cos \left (f x +e \right ) a \right )^{m} \left (b \csc \left (f x +e \right )\right )^{n}d x\]
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\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=\int { \left (a \cos \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=\int \left (a \cos {\left (e + f x \right )}\right )^{m} \left (b \csc {\left (e + f x \right )}\right )^{n}\, dx \]
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\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=\int { \left (a \cos \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=\int { \left (a \cos \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx=\int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n \,d x \]
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