Integrand size = 14, antiderivative size = 89 \[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\frac {2 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-5 m),\frac {1}{4} (6-5 m),\sin ^2(a+b x)\right ) \sin ^{1-2 m}(a+b x)}{b c^2 (2-5 m) \sqrt {\cos ^2(a+b x)} \sqrt {c \sin ^m(a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3287, 2722} \[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\frac {2 \cos (a+b x) \sin ^{1-2 m}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-5 m),\frac {1}{4} (6-5 m),\sin ^2(a+b x)\right )}{b c^2 (2-5 m) \sqrt {\cos ^2(a+b x)} \sqrt {c \sin ^m(a+b x)}} \]
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Rule 2722
Rule 3287
Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^{\frac {m}{2}}(a+b x) \int \sin ^{-\frac {5 m}{2}}(a+b x) \, dx}{c^2 \sqrt {c \sin ^m(a+b x)}} \\ & = \frac {2 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-5 m),\frac {1}{4} (6-5 m),\sin ^2(a+b x)\right ) \sin ^{1-2 m}(a+b x)}{b c^2 (2-5 m) \sqrt {\cos ^2(a+b x)} \sqrt {c \sin ^m(a+b x)}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\frac {\sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-5 m),\frac {1}{4} (6-5 m),\sin ^2(a+b x)\right ) \tan (a+b x)}{\left (b-\frac {5 b m}{2}\right ) \left (c \sin ^m(a+b x)\right )^{5/2}} \]
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\[\int \frac {1}{{\left (c \left (\sin ^{m}\left (b x +a \right )\right )\right )}^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\int \frac {1}{\left (c \sin ^{m}{\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )^{m}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )^{m}\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c \sin ^m(a+b x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,{\sin \left (a+b\,x\right )}^m\right )}^{5/2}} \,d x \]
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