Integrand size = 14, antiderivative size = 89 \[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=-\frac {2 \cos ^{1-m}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-3 m),\frac {3 (2-m)}{4},\cos ^2(a+b x)\right ) \sin (a+b x)}{b c (2-3 m) \sqrt {c \cos ^m(a+b x)} \sqrt {\sin ^2(a+b x)}} \]
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Time = 0.05 (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} \[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=-\frac {2 \sin (a+b x) \cos ^{1-m}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-3 m),\frac {3 (2-m)}{4},\cos ^2(a+b x)\right )}{b c (2-3 m) \sqrt {\sin ^2(a+b x)} \sqrt {c \cos ^m(a+b x)}} \]
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Rule 2722
Rule 3287
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^{\frac {m}{2}}(a+b x) \int \cos ^{-\frac {3 m}{2}}(a+b x) \, dx}{c \sqrt {c \cos ^m(a+b x)}} \\ & = -\frac {2 \cos ^{1-m}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-3 m),\frac {3 (2-m)}{4},\cos ^2(a+b x)\right ) \sin (a+b x)}{b c (2-3 m) \sqrt {c \cos ^m(a+b x)} \sqrt {\sin ^2(a+b x)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=-\frac {\cot (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2-3 m),-\frac {3}{4} (-2+m),\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{\left (b-\frac {3 b m}{2}\right ) \left (c \cos ^m(a+b x)\right )^{3/2}} \]
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\[\int \frac {1}{{\left (c \left (\cos ^{m}\left (b x +a \right )\right )\right )}^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=\int \frac {1}{\left (c \cos ^{m}{\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (c \cos \left (b x + a\right )^{m}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (c \cos \left (b x + a\right )^{m}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c \cos ^m(a+b x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,{\cos \left (a+b\,x\right )}^m\right )}^{3/2}} \,d x \]
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