Integrand size = 21, antiderivative size = 56 \[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\frac {3 b \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3} \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {16, 2722} \[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\frac {3 b \sin (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\cos ^2(c+d x)\right )}{5 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}} \]
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Rule 16
Rule 2722
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {1}{(b \cos (c+d x))^{8/3}} \, dx \\ & = \frac {3 b \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\frac {3 b^2 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{5 d (b \cos (c+d x))^{8/3}} \]
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\[\int \frac {\sec ^{2}\left (d x +c \right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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