Integrand size = 10, antiderivative size = 51 \[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=-\frac {3 \sqrt [3]{\cos (a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\cos ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt {\sin ^2(a+b x)}} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722} \[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=-\frac {3 \sin (a+b x) \sqrt [3]{\cos (a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\cos ^2(a+b x)\right )}{b \sqrt {\sin ^2(a+b x)}} \]
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Rule 2722
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt [3]{\cos (a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\cos ^2(a+b x)\right ) \sin (a+b x)}{b \sqrt {\sin ^2(a+b x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=-\frac {3 \sqrt [3]{\cos (a+b x)} \csc (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{b} \]
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\[\int \frac {1}{\cos \left (b x +a \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=\int { \frac {1}{\cos \left (b x + a\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=\int \frac {1}{\cos ^{\frac {2}{3}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=\int { \frac {1}{\cos \left (b x + a\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=\int { \frac {1}{\cos \left (b x + a\right )^{\frac {2}{3}}} \,d x } \]
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Time = 15.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\cos ^{\frac {2}{3}}(a+b x)} \, dx=-\frac {3\,{\cos \left (a+b\,x\right )}^{1/3}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{6},\frac {1}{2};\ \frac {7}{6};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \]
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