Integrand size = 19, antiderivative size = 58 \[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {3 (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{8 b^2 d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {16, 2722} \[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {3 \sin (c+d x) (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right )}{8 b^2 d \sqrt {\sin ^2(c+d x)}} \]
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Rule 16
Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{5/3} \, dx}{b} \\ & = -\frac {3 (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{8 b^2 d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=-\frac {3 (b \cos (c+d x))^{5/3} \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{8 b d} \]
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\[\int \cos \left (d x +c \right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}}d x\]
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\[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=\text {Timed out} \]
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\[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right ) \,d x } \]
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\[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{2/3} \, dx=\int \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \]
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