Integrand size = 39, antiderivative size = 148 \[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 d}-\frac {3 (5 A+2 C) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{10 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{5/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {16, 3102, 2827, 2722} \[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=-\frac {3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right )}{10 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(c+d x)\right )}{5 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 d} \]
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Rule 16
Rule 2722
Rule 2827
Rule 3102
Rubi steps \begin{align*} \text {integral}& = b \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx \\ & = \frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 d}+\frac {3}{5} \int \frac {\frac {1}{3} b (5 A+2 C)+\frac {5}{3} b B \cos (c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx \\ & = \frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 d}+B \int (b \cos (c+d x))^{2/3} \, dx+\frac {1}{5} (b (5 A+2 C)) \int \frac {1}{\sqrt [3]{b \cos (c+d x)}} \, dx \\ & = \frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 d}-\frac {3 (5 A+2 C) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{10 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{5/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {b \left (-3 (5 A+2 C) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}-6 B \cos (c+d x) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}+3 C \sin (2 (c+d x))\right )}{10 d \sqrt [3]{b \cos (c+d x)}} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \sec \left (d x +c \right )d x\]
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\[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \]
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\[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right ) \,d x } \]
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\[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{2/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\cos \left (c+d\,x\right )} \,d x \]
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