Integrand size = 23, antiderivative size = 76 \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=-\frac {b (a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a f (1+m)} \]
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Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2667, 2656} \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=-\frac {b \sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)} (a \cos (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(e+f x)\right )}{a f (m+1)} \]
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Rule 2656
Rule 2667
Rubi steps \begin{align*} \text {integral}& = \left (b^2 \sqrt {b \csc (e+f x)} \sqrt {b \sin (e+f x)}\right ) \int \frac {(a \cos (e+f x))^m}{(b \sin (e+f x))^{3/2}} \, dx \\ & = -\frac {b (a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a f (1+m)} \\ \end{align*}
Time = 9.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.24 \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=\frac {2 a b (a \cos (e+f x))^{-1+m} \left (-\cot ^2(e+f x)\right )^{\frac {1-m}{2}} \sqrt {b \csc (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} (3-2 m),\frac {1-m}{2},\frac {1}{4} (7-2 m),\csc ^2(e+f x)\right )}{f (-3+2 m)} \]
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\[\int \left (\cos \left (f x +e \right ) a \right )^{m} \left (b \csc \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
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\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \cos \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \cos \left (f x + e\right )\right )^{m} \,d x } \]
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\[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{\frac {3}{2}} \left (a \cos \left (f x + e\right )\right )^{m} \,d x } \]
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Timed out. \[ \int (a \cos (e+f x))^m (b \csc (e+f x))^{3/2} \, dx=\int {\left (a\,\cos \left (e+f\,x\right )\right )}^m\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
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