Integrand size = 21, antiderivative size = 82 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 82, 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 = {20, 2722} \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {3 \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+5),\frac {1}{6} (3 m+11),\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt {\sin ^2(c+d x)}} \]
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Rule 20
Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {(b \cos (c+d x))^{2/3} \int \cos ^{\frac {2}{3}+m}(c+d x) \, dx}{\cos ^{\frac {2}{3}}(c+d x)} \\ & = -\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {\cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {5}{3}+m\right ),\frac {1}{2} \left (\frac {11}{3}+m\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {5}{3}+m\right )} \]
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\[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}}d x\]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}} \cos ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \, dx=\int {\cos \left (c+d\,x\right )}^m\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \]
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