Integrand size = 12, antiderivative size = 56 \[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(a+b x)\right ) \sin (a+b x)}{b c \sqrt [3]{c \cos (a+b x)} \sqrt {\sin ^2(a+b x)}} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\frac {3 \sin (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(a+b x)\right )}{b c \sqrt {\sin ^2(a+b x)} \sqrt [3]{c \cos (a+b x)}} \]
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Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(a+b x)\right ) \sin (a+b x)}{b c \sqrt [3]{c \cos (a+b x)} \sqrt {\sin ^2(a+b x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\frac {3 \cot (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{b (c \cos (a+b x))^{4/3}} \]
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\[\int \frac {1}{\left (c \cos \left (b x +a \right )\right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\int { \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\int \frac {1}{\left (c \cos {\left (a + b x \right )}\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\int { \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\int { \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(c \cos (a+b x))^{4/3}} \, dx=\int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{4/3}} \,d x \]
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