Optimal. Leaf size=162 \[ -\frac {388\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{455 d (\sin (c+d x)+1)^{5/6}}-\frac {3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{65 a d}-\frac {72 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{455 d} \]
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Rubi [A] time = 0.28, antiderivative size = 162, 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 = {2783, 2968, 3023, 2751, 2652, 2651} \[ -\frac {388\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{455 d (\sin (c+d x)+1)^{5/6}}-\frac {3 \sin ^2(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a \sin (c+d x)+a)^{7/3}}{65 a d}-\frac {72 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{455 d} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rule 2751
Rule 2783
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \sin ^3(c+d x) (a+a \sin (c+d x))^{4/3} \, dx &=-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}+\frac {3 \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \left (2 a+\frac {4}{3} a \sin (c+d x)\right ) \, dx}{13 a}\\ &=-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}+\frac {3 \int (a+a \sin (c+d x))^{4/3} \left (2 a \sin (c+d x)+\frac {4}{3} a \sin ^2(c+d x)\right ) \, dx}{13 a}\\ &=-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d}+\frac {9 \int (a+a \sin (c+d x))^{4/3} \left (\frac {28 a^2}{9}+\frac {16}{3} a^2 \sin (c+d x)\right ) \, dx}{130 a^2}\\ &=-\frac {72 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{455 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d}+\frac {194}{455} \int (a+a \sin (c+d x))^{4/3} \, dx\\ &=-\frac {72 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{455 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d}+\frac {\left (194 a \sqrt [3]{a+a \sin (c+d x)}\right ) \int (1+\sin (c+d x))^{4/3} \, dx}{455 \sqrt [3]{1+\sin (c+d x)}}\\ &=-\frac {388\ 2^{5/6} a \cos (c+d x) \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{455 d (1+\sin (c+d x))^{5/6}}-\frac {72 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{455 d}-\frac {3 \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^{4/3}}{13 d}-\frac {6 \cos (c+d x) (a+a \sin (c+d x))^{7/3}}{65 a d}\\ \end {align*}
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Mathematica [C] time = 2.64, size = 373, normalized size = 2.30 \[ \frac {(a (\sin (c+d x)+1))^{4/3} \left (-\frac {3}{40} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right ) (278 \sin (2 (c+d x))-35 \sin (4 (c+d x))+790 \cos (c+d x)-98 \cos (3 (c+d x))-1940)+\frac {291 (-1)^{3/4} e^{-\frac {3}{2} i (c+d x)} \left (e^{i (c+d x)}+i\right ) \left (-2 \left (1+i e^{-i (c+d x)}\right )^{2/3} \left (1+e^{2 i (c+d x)}\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\sin ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )\right )+5 i \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-i e^{-i (c+d x)}\right ) \sqrt {2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-i e^{-i (c+d x)}\right ) \sqrt {\cos ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )}\right )}{20 \sqrt {2} \left (1+i e^{-i (c+d x)}\right )^{2/3} \sqrt {i e^{-i (c+d x)} \left (e^{i (c+d x)}-i\right )^2}}\right )}{91 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) + a\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.59, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}\, dx \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \sin \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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