Integrand size = 31, antiderivative size = 89 \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 d}-\frac {3 (7 A+4 C) (b \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{28 d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {16, 3093, 2722} \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 d}-\frac {3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right )}{28 d \sqrt {\sin ^2(c+d x)}} \]
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Rule 16
Rule 2722
Rule 3093
Rubi steps \begin{align*} \text {integral}& = b \int \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 d}+\frac {1}{7} (b (7 A+4 C)) \int \sqrt [3]{b \cos (c+d x)} \, dx \\ & = \frac {3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 d}-\frac {3 (7 A+4 C) (b \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{28 d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=-\frac {3 b \sqrt [3]{b \cos (c+d x)} \cot (c+d x) \left (5 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right )+2 C \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{20 d} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{\frac {4}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \sec \left (d x +c \right )d x\]
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\[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \]
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\[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right ) \,d x } \]
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\[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}}{\cos \left (c+d\,x\right )} \,d x \]
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