Integrand size = 12, antiderivative size = 58 \[ \int \sqrt [3]{b \cos (c+d x)} \, dx=-\frac {3 (b \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{4 b d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int \sqrt [3]{b \cos (c+d x)} \, dx=-\frac {3 \sin (c+d x) (b \cos (c+d x))^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right )}{4 b d \sqrt {\sin ^2(c+d x)}} \]
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Rule 2722
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (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)}{4 b d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \sqrt [3]{b \cos (c+d x)} \, dx=-\frac {3 \sqrt [3]{b \cos (c+d x)} \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{4 d} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}}d x\]
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\[ \int \sqrt [3]{b \cos (c+d x)} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{b \cos (c+d x)} \, dx=\int \sqrt [3]{b \cos {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt [3]{b \cos (c+d x)} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{b \cos (c+d x)} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \,d x } \]
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Timed out. \[ \int \sqrt [3]{b \cos (c+d x)} \, dx=\int {\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \]
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