Integrand size = 23, antiderivative size = 78 \[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=-\frac {(a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a b f (1+m)} \]
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Time = 0.11 (sec) , antiderivative size = 78, 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 = {2666, 2656} \[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=-\frac {\sqrt [4]{\sin ^2(e+f x)} \sqrt {b \csc (e+f x)} (a \cos (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(e+f x)\right )}{a b f (m+1)} \]
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Rule 2656
Rule 2666
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {b \csc (e+f x)} \sqrt {b \sin (e+f x)}\right ) \int (a \cos (e+f x))^m \sqrt {b \sin (e+f x)} \, dx}{b^2} \\ & = -\frac {(a \cos (e+f x))^{1+m} \sqrt {b \csc (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(e+f x)\right ) \sqrt [4]{\sin ^2(e+f x)}}{a b f (1+m)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 28.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.88 \[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\frac {14 b \operatorname {AppellF1}\left (\frac {3}{4},-m,\frac {3}{2}+m,\frac {7}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (a \cos (e+f x))^m}{3 f (b \csc (e+f x))^{3/2} \left (7 \operatorname {AppellF1}\left (\frac {3}{4},-m,\frac {3}{2}+m,\frac {7}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left (2 m \operatorname {AppellF1}\left (\frac {7}{4},1-m,\frac {3}{2}+m,\frac {11}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(3+2 m) \operatorname {AppellF1}\left (\frac {7}{4},-m,\frac {5}{2}+m,\frac {11}{4},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \frac {\left (\cos \left (f x +e \right ) a \right )^{m}}{\sqrt {b \csc \left (f x +e \right )}}d x\]
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\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \csc \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int \frac {\left (a \cos {\left (e + f x \right )}\right )^{m}}{\sqrt {b \csc {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \csc \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int { \frac {\left (a \cos \left (f x + e\right )\right )^{m}}{\sqrt {b \csc \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a \cos (e+f x))^m}{\sqrt {b \csc (e+f x)}} \, dx=\int \frac {{\left (a\,\cos \left (e+f\,x\right )\right )}^m}{\sqrt {\frac {b}{\sin \left (e+f\,x\right )}}} \,d x \]
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