Integrand size = 14, antiderivative size = 82 \[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=\frac {\cos (c+d x) \left (a (b \csc (c+d x))^p\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1-n p),\frac {1}{2} (3-n p),\sin ^2(c+d x)\right ) \sin (c+d x)}{d (1-n p) \sqrt {\cos ^2(c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4208, 3857, 2722} \[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=\frac {\sin (c+d x) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1-n p),\frac {1}{2} (3-n p),\sin ^2(c+d x)\right ) \left (a (b \csc (c+d x))^p\right )^n}{d (1-n p) \sqrt {\cos ^2(c+d x)}} \]
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Rule 2722
Rule 3857
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \left ((b \csc (c+d x))^{-n p} \left (a (b \csc (c+d x))^p\right )^n\right ) \int (b \csc (c+d x))^{n p} \, dx \\ & = \left (\left (a (b \csc (c+d x))^p\right )^n \left (\frac {\sin (c+d x)}{b}\right )^{n p}\right ) \int \left (\frac {\sin (c+d x)}{b}\right )^{-n p} \, dx \\ & = \frac {\cos (c+d x) \left (a (b \csc (c+d x))^p\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1-n p),\frac {1}{2} (3-n p),\sin ^2(c+d x)\right ) \sin (c+d x)}{d (1-n p) \sqrt {\cos ^2(c+d x)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=-\frac {\cos (c+d x) \left (a (b \csc (c+d x))^p\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {3}{2},\cos ^2(c+d x)\right ) \sin (c+d x) \sin ^2(c+d x)^{\frac {1}{2} (-1+n p)}}{d} \]
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\[\int \left (a \left (b \csc \left (d x +c \right )\right )^{p}\right )^{n}d x\]
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\[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=\int { \left (\left (b \csc \left (d x + c\right )\right )^{p} a\right )^{n} \,d x } \]
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\[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=\int \left (a \left (b \csc {\left (c + d x \right )}\right )^{p}\right )^{n}\, dx \]
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\[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=\int { \left (\left (b \csc \left (d x + c\right )\right )^{p} a\right )^{n} \,d x } \]
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\[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=\int { \left (\left (b \csc \left (d x + c\right )\right )^{p} a\right )^{n} \,d x } \]
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Timed out. \[ \int \left (a (b \csc (c+d x))^p\right )^n \, dx=\int {\left (a\,{\left (\frac {b}{\sin \left (c+d\,x\right )}\right )}^p\right )}^n \,d x \]
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