Integrand size = 23, antiderivative size = 184 \[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\frac {4 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2(c+d x)\right ) \sqrt [3]{\sin (c+d x)}}{9 a^2 d \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(c+d x)\right ) \sin ^{\frac {4}{3}}(c+d x)}{36 a^2 d \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{9 a^2 d (1+\sin (c+d x))}-\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{3 d (a+a \sin (c+d x))^2} \]
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Time = 0.15 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2843, 3057, 2827, 2722} \[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\frac {4 \sqrt [3]{\sin (c+d x)} \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2(c+d x)\right )}{9 a^2 d \sqrt {\cos ^2(c+d x)}}-\frac {\sin ^{\frac {4}{3}}(c+d x) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(c+d x)\right )}{36 a^2 d \sqrt {\cos ^2(c+d x)}}-\frac {\sqrt [3]{\sin (c+d x)} \cos (c+d x)}{9 a^2 d (\sin (c+d x)+1)}-\frac {\sqrt [3]{\sin (c+d x)} \cos (c+d x)}{3 d (a \sin (c+d x)+a)^2} \]
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Rule 2722
Rule 2827
Rule 2843
Rule 3057
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{3 d (a+a \sin (c+d x))^2}+\frac {\int \frac {\frac {a}{3}+\frac {2}{3} a \sin (c+d x)}{\sin ^{\frac {2}{3}}(c+d x) (a+a \sin (c+d x))} \, dx}{3 a^2} \\ & = -\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{9 a^2 d (1+\sin (c+d x))}-\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{3 d (a+a \sin (c+d x))^2}+\frac {\int \frac {\frac {4 a^2}{9}-\frac {1}{9} a^2 \sin (c+d x)}{\sin ^{\frac {2}{3}}(c+d x)} \, dx}{3 a^4} \\ & = -\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{9 a^2 d (1+\sin (c+d x))}-\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{3 d (a+a \sin (c+d x))^2}-\frac {\int \sqrt [3]{\sin (c+d x)} \, dx}{27 a^2}+\frac {4 \int \frac {1}{\sin ^{\frac {2}{3}}(c+d x)} \, dx}{27 a^2} \\ & = \frac {4 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2(c+d x)\right ) \sqrt [3]{\sin (c+d x)}}{9 a^2 d \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\sin ^2(c+d x)\right ) \sin ^{\frac {4}{3}}(c+d x)}{36 a^2 d \sqrt {\cos ^2(c+d x)}}-\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{9 a^2 d (1+\sin (c+d x))}-\frac {\cos (c+d x) \sqrt [3]{\sin (c+d x)}}{3 d (a+a \sin (c+d x))^2} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\frac {\sec ^3(c+d x) \sqrt [3]{\sin (c+d x)} \left (80 \cos ^2(c+d x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2(c+d x)\right )+27 \cos ^2(c+d x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{2},\frac {5}{3},\sin ^2(c+d x)\right ) \sin (c+d x)+4 (-25+5 \cos (2 (c+d x))+27 \sin (c+d x))\right )}{180 a^2 d} \]
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\[\int \frac {\sin ^{\frac {1}{3}}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{2}}d x\]
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\[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{\frac {1}{3}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\frac {\int \frac {\sqrt [3]{\sin {\left (c + d x \right )}}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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\[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{\frac {1}{3}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{\frac {1}{3}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^{1/3}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]
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