Integrand size = 21, antiderivative size = 58 \[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\frac {3 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 58, 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} \[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\frac {3 b^2 \sin (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]
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Rule 16
Rule 2722
Rubi steps \begin{align*} \text {integral}& = b^3 \int \frac {1}{(b \cos (c+d x))^{7/3}} \, dx \\ & = \frac {3 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\frac {3 (b \cos (c+d x))^{2/3} \csc (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right ) \sec ^2(c+d x) \sqrt {\sin ^2(c+d x)}}{4 d} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}} \left (\sec ^{3}\left (d x +c \right )\right )d x\]
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\[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\text {Timed out} \]
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\[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{3} \,d x } \]
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\[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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