Optimal. Leaf size=161 \[ -\frac {63 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{220 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3}}{11 d}-\frac {67 \cos (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{55\ 2^{5/6} d (1+\sin (c+d x))^{7/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{44 a d} \]
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Rubi [A]
time = 0.20, 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 {67 \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{55\ 2^{5/6} d (\sin (c+d x)+1)^{7/6}}-\frac {3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{11 d}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{5/3}}{44 a d}-\frac {63 \cos (c+d x) (a \sin (c+d x)+a)^{2/3}}{220 d} \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 \sin ^3(c+d x) (a+a \sin (c+d x))^{2/3} \, dx &=-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3}}{11 d}+\frac {3 \int \sin (c+d x) \left (2 a+\frac {2}{3} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{2/3} \, dx}{11 a}\\ &=-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3}}{11 d}+\frac {3 \int (a+a \sin (c+d x))^{2/3} \left (2 a \sin (c+d x)+\frac {2}{3} a \sin ^2(c+d x)\right ) \, dx}{11 a}\\ &=-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3}}{11 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{44 a d}+\frac {9 \int (a+a \sin (c+d x))^{2/3} \left (\frac {10 a^2}{9}+\frac {14}{3} a^2 \sin (c+d x)\right ) \, dx}{88 a^2}\\ &=-\frac {63 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{220 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3}}{11 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{44 a d}+\frac {67}{220} \int (a+a \sin (c+d x))^{2/3} \, dx\\ &=-\frac {63 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{220 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3}}{11 d}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{44 a d}+\frac {\left (67 (a+a \sin (c+d x))^{2/3}\right ) \int (1+\sin (c+d x))^{2/3} \, dx}{220 (1+\sin (c+d x))^{2/3}}\\ &=-\frac {63 \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{220 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{2/3}}{11 d}-\frac {67 \cos (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^{2/3}}{55\ 2^{5/6} d (1+\sin (c+d x))^{7/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{5/3}}{44 a d}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 160, normalized size = 0.99 \begin {gather*} \frac {3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a (1+\sin (c+d x)))^{2/3} \left (67 \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)} (-144+25 \cos (2 (c+d x))-92 \sin (c+d x)+10 \sin (3 (c+d x)))\right )}{440 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {1-\sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \left (\sin ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}\, dx\]
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 {\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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