Integrand size = 33, antiderivative size = 95 \[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C (b \cos (c+d x))^{10/3} \sin (c+d x)}{13 b^3 d}-\frac {3 (13 A+10 C) (b \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{130 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {16, 3093, 2722} \[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b^3 d}-\frac {3 (13 A+10 C) \sin (c+d x) (b \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right )}{130 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Rule 16
Rule 2722
Rule 3093
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{7/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2} \\ & = \frac {3 C (b \cos (c+d x))^{10/3} \sin (c+d x)}{13 b^3 d}+\frac {(13 A+10 C) \int (b \cos (c+d x))^{7/3} \, dx}{13 b^2} \\ & = \frac {3 C (b \cos (c+d x))^{10/3} \sin (c+d x)}{13 b^3 d}-\frac {3 (13 A+10 C) (b \cos (c+d x))^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{130 b^3 d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {3 \sqrt [3]{b \cos (c+d x)} \cot (c+d x) \left (8 A \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3},\frac {8}{3},\cos ^2(c+d x)\right )+5 C \cos ^4(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {8}{3},\frac {11}{3},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{80 d} \]
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\[\int \left (\cos ^{2}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
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\[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{2} \,d x } \]
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\[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \]
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