Integrand size = 14, antiderivative size = 66 \[ \int (a+a \cos (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} (a+a \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}} \]
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Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2731, 2730} \[ \int (a+a \cos (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{d (\cos (c+d x)+1)^{7/6}} \]
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Rule 2730
Rule 2731
Rubi steps \begin{align*} \text {integral}& = \frac {(a+a \cos (c+d x))^{2/3} \int (1+\cos (c+d x))^{2/3} \, dx}{(1+\cos (c+d x))^{2/3}} \\ & = \frac {2 \sqrt [6]{2} (a+a \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{d (1+\cos (c+d x))^{7/6}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int (a+a \cos (c+d x))^{2/3} \, dx=-\frac {6 (a (1+\cos (c+d x)))^{2/3} \cot \left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{7 d} \]
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\[\int \left (a +\cos \left (d x +c \right ) a \right )^{\frac {2}{3}}d x\]
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\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int \left (a \cos {\left (c + d x \right )} + a\right )^{\frac {2}{3}}\, dx \]
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\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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\[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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Timed out. \[ \int (a+a \cos (c+d x))^{2/3} \, dx=\int {\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \]
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