Integrand size = 14, antiderivative size = 65 \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\frac {\sqrt [6]{1+\cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{\sqrt [6]{2} d (a+a \cos (c+d x))^{2/3}} \]
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Time = 0.04 (sec) , antiderivative size = 65, 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} \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\frac {\sin (c+d x) \sqrt [6]{\cos (c+d x)+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right )}{\sqrt [6]{2} d (a \cos (c+d x)+a)^{2/3}} \]
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Rule 2730
Rule 2731
Rubi steps \begin{align*} \text {integral}& = \frac {(1+\cos (c+d x))^{2/3} \int \frac {1}{(1+\cos (c+d x))^{2/3}} \, dx}{(a+a \cos (c+d x))^{2/3}} \\ & = \frac {\sqrt [6]{1+\cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x))\right ) \sin (c+d x)}{\sqrt [6]{2} d (a+a \cos (c+d x))^{2/3}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\frac {6 \cot \left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}{d (a (1+\cos (c+d x)))^{2/3}} \]
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\[\int \frac {1}{\left (a +\cos \left (d x +c \right ) a \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (a \cos {\left (c + d x \right )} + a\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{2/3}} \, dx=\int \frac {1}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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