Integrand size = 33, antiderivative size = 146 \[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{d (7+3 m)}-\frac {3 (C (4+3 m)+A (7+3 m)) \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {1}{6} (10+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (7+3 m) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {20, 3093, 2722} \[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+1}(c+d x)}{d (3 m+7)}-\frac {3 \left (\frac {A}{3 m+4}+\frac {C}{3 m+7}\right ) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+4),\frac {1}{6} (3 m+10),\cos ^2(c+d x)\right )}{d \sqrt {\sin ^2(c+d x)}} \]
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Rule 20
Rule 2722
Rule 3093
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{b \cos (c+d x)} \int \cos ^{\frac {1}{3}+m}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{\sqrt [3]{\cos (c+d x)}} \\ & = \frac {3 C \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{d (7+3 m)}+\frac {\left (\left (C \left (\frac {4}{3}+m\right )+A \left (\frac {7}{3}+m\right )\right ) \sqrt [3]{b \cos (c+d x)}\right ) \int \cos ^{\frac {1}{3}+m}(c+d x) \, dx}{\left (\frac {7}{3}+m\right ) \sqrt [3]{\cos (c+d x)}} \\ & = \frac {3 C \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{d (7+3 m)}-\frac {3 (C (4+3 m)+A (7+3 m)) \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {1}{6} (10+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (7+3 m) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {3 \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \csc (c+d x) \left (C (4+3 m) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{3}+\frac {m}{2},\frac {8}{3}+\frac {m}{2},\cos ^2(c+d x)\right )+A (10+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {5}{3}+\frac {m}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (4+3 m) (10+3 m)} \]
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\[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
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\[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \sqrt [3]{b \cos {\left (c + d x \right )}} \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \]
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