Integrand size = 21, antiderivative size = 91 \[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=\frac {a \cos (e+f x) (a \csc (e+f x))^{-1+m} (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1-m-n),\frac {1}{2} (3-m-n),\sin ^2(e+f x)\right )}{f (1-m-n) \sqrt {\cos ^2(e+f x)}} \]
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Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {20, 3857, 2722} \[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=\frac {a \cos (e+f x) (a \csc (e+f x))^{m-1} (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-m-n+1),\frac {1}{2} (-m-n+3),\sin ^2(e+f x)\right )}{f (-m-n+1) \sqrt {\cos ^2(e+f x)}} \]
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Rule 20
Rule 2722
Rule 3857
Rubi steps \begin{align*} \text {integral}& = \left ((a \csc (e+f x))^{-n} (b \csc (e+f x))^n\right ) \int (a \csc (e+f x))^{m+n} \, dx \\ & = \left ((a \csc (e+f x))^m (b \csc (e+f x))^n \left (\frac {\sin (e+f x)}{a}\right )^{m+n}\right ) \int \left (\frac {\sin (e+f x)}{a}\right )^{-m-n} \, dx \\ & = \frac {\cos (e+f x) (a \csc (e+f x))^m (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1-m-n),\frac {1}{2} (3-m-n),\sin ^2(e+f x)\right ) \sin (e+f x)}{f (1-m-n) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.85 \[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=-\frac {\cos (e+f x) (a \csc (e+f x))^m (b \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {3}{2},\cos ^2(e+f x)\right ) \sin (e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-1+m+n)}}{f} \]
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\[\int \left (a \csc \left (f x +e \right )\right )^{m} \left (b \csc \left (f x +e \right )\right )^{n}d x\]
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\[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=\int { \left (a \csc \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=\int \left (a \csc {\left (e + f x \right )}\right )^{m} \left (b \csc {\left (e + f x \right )}\right )^{n}\, dx \]
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\[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=\int { \left (a \csc \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=\int { \left (a \csc \left (f x + e\right )\right )^{m} \left (b \csc \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx=\int {\left (\frac {a}{\sin \left (e+f\,x\right )}\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n \,d x \]
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