Optimal. Leaf size=161 \[ \frac{37 \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{40\ 2^{5/6} d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{8 d \sqrt [3]{a \sin (c+d x)+a}}+\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 a d}-\frac{99 \cos (c+d x)}{80 d \sqrt [3]{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.252884, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2783, 2968, 3023, 2751, 2652, 2651} \[ \frac{37 \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{40\ 2^{5/6} d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}-\frac{3 \sin ^2(c+d x) \cos (c+d x)}{8 d \sqrt [3]{a \sin (c+d x)+a}}+\frac{3 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{40 a d}-\frac{99 \cos (c+d x)}{80 d \sqrt [3]{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2783
Rule 2968
Rule 3023
Rule 2751
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx &=-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac{3 \int \frac{\sin (c+d x) \left (2 a-\frac{1}{3} a \sin (c+d x)\right )}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{8 a}\\ &=-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac{3 \int \frac{2 a \sin (c+d x)-\frac{1}{3} a \sin ^2(c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{8 a}\\ &=-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}+\frac{9 \int \frac{-\frac{2 a^2}{9}+\frac{11}{3} a^2 \sin (c+d x)}{\sqrt [3]{a+a \sin (c+d x)}} \, dx}{40 a^2}\\ &=-\frac{99 \cos (c+d x)}{80 d \sqrt [3]{a+a \sin (c+d x)}}-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}-\frac{37}{80} \int \frac{1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx\\ &=-\frac{99 \cos (c+d x)}{80 d \sqrt [3]{a+a \sin (c+d x)}}-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}-\frac{\left (37 \sqrt [3]{1+\sin (c+d x)}\right ) \int \frac{1}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{80 \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{99 \cos (c+d x)}{80 d \sqrt [3]{a+a \sin (c+d x)}}-\frac{3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac{37 \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{40\ 2^{5/6} d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}+\frac{3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d}\\ \end{align*}
Mathematica [A] time = 0.45825, size = 110, normalized size = 0.68 \[ \frac{3 \cos (c+d x) \left (\sqrt{1-\sin (c+d x)} (2 \sin (c+d x)+5 \cos (2 (c+d x))-36)-37 \sqrt{2} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )\right )\right )}{80 d \sqrt{1-\sin (c+d x)} \sqrt [3]{a (\sin (c+d x)+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}{\frac{1}{\sqrt [3]{a+a\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 )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{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{{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}, 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 )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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