Optimal. Leaf size=184 \[ -\frac{\sin ^{\frac{4}{3}}(c+d x) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(c+d x)\right )}{36 a^2 d \sqrt{\cos ^2(c+d x)}}+\frac{4 \sqrt [3]{\sin (c+d x)} \cos (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(c+d x)\right )}{9 a^2 d \sqrt{\cos ^2(c+d x)}}-\frac{\sqrt [3]{\sin (c+d x)} \cos (c+d x)}{9 a^2 d (\sin (c+d x)+1)}-\frac{\sqrt [3]{\sin (c+d x)} \cos (c+d x)}{3 d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.213211, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2764, 2978, 2748, 2643} \[ -\frac{\sin ^{\frac{4}{3}}(c+d x) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(c+d x)\right )}{36 a^2 d \sqrt{\cos ^2(c+d x)}}+\frac{4 \sqrt [3]{\sin (c+d x)} \cos (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(c+d x)\right )}{9 a^2 d \sqrt{\cos ^2(c+d x)}}-\frac{\sqrt [3]{\sin (c+d x)} \cos (c+d x)}{9 a^2 d (\sin (c+d x)+1)}-\frac{\sqrt [3]{\sin (c+d x)} \cos (c+d x)}{3 d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2764
Rule 2978
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{\sin (c+d x)}}{(a+a \sin (c+d x))^2} \, dx &=-\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) \, _2F_1\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) \, _2F_1\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*}
Mathematica [A] time = 0.403805, size = 121, normalized size = 0.66 \[ \frac{\sqrt [3]{\sin (c+d x)} \sec ^3(c+d x) \left (80 \cos ^2(c+d x)^{3/2} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(c+d x)\right )+27 \sin (c+d x) \cos ^2(c+d x)^{3/2} \, _2F_1\left (\frac{2}{3},\frac{5}{2};\frac{5}{3};\sin ^2(c+d x)\right )+4 (27 \sin (c+d x)+5 \cos (2 (c+d x))-25)\right )}{180 a^2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.219, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{2}}\sqrt [3]{\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )^{\frac{1}{3}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sin \left (d x + c\right )^{\frac{1}{3}}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )^{\frac{1}{3}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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