Optimal. Leaf size=161 \[ -\frac {99 \cos (c+d x)}{80 d \sqrt [3]{a+a \sin (c+d x)}}-\frac {3 \cos (c+d x) \sin ^2(c+d x)}{8 d \sqrt [3]{a+a \sin (c+d x)}}+\frac {37 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{40\ 2^{5/6} d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}+\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{40 a d} \]
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Rubi [A]
time = 0.19, antiderivative size = 161, normalized size of antiderivative = 1.00, 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 = {2862, 3047,
3102, 2830, 2731, 2730} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rule 2731
Rule 2830
Rule 2862
Rule 3047
Rule 3102
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*}
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Mathematica [A]
time = 0.31, size = 110, normalized size = 0.68 \begin {gather*} \frac {3 \cos (c+d x) \left (-37 \sqrt {2} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+\sqrt {1-\sin (c+d x)} (-36+5 \cos (2 (c+d x))+2 \sin (c+d x))\right )}{80 d \sqrt {1-\sin (c+d x)} \sqrt [3]{a (1+\sin (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.53, size = 0, normalized size = 0.00 \[\int \frac {\sin ^{3}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\sin \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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