Optimal. Leaf size=97 \[ -\frac {8\ 2^{5/6} a \cos (c+d x) \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{7 d (1+\sin (c+d x))^{5/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2830, 2731,
2730} \begin {gather*} -\frac {8\ 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 )}{7 d (\sin (c+d x)+1)^{5/6}}-\frac {3 \cos (c+d x) (a \sin (c+d x)+a)^{4/3}}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rule 2731
Rule 2830
Rubi steps
\begin {align*} \int \sin (c+d x) (a+a \sin (c+d x))^{4/3} \, dx &=-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d}+\frac {4}{7} \int (a+a \sin (c+d x))^{4/3} \, dx\\ &=-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d}+\frac {\left (4 a \sqrt [3]{a+a \sin (c+d x)}\right ) \int (1+\sin (c+d x))^{4/3} \, dx}{7 \sqrt [3]{1+\sin (c+d x)}}\\ &=-\frac {8\ 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)}}{7 d (1+\sin (c+d x))^{5/6}}-\frac {3 \cos (c+d x) (a+a \sin (c+d x))^{4/3}}{7 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 8.75, size = 351, normalized size = 3.62 \begin {gather*} \frac {(a (1+\sin (c+d x)))^{4/3} \left (\frac {3 (-1)^{3/4} e^{-\frac {3}{2} i (c+d x)} \left (i+e^{i (c+d x)}\right ) \left (20 e^{i (c+d x)} \sqrt {\cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-i e^{-i (c+d x)}\right )-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+\pi +2 d x)\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)}\right )}{2 \sqrt {2} \left (1+i e^{-i (c+d x)}\right )^{2/3} \sqrt {i e^{-i (c+d x)} \left (-i+e^{i (c+d x)}\right )^2}}-\frac {3}{2} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (-10+4 \cos (c+d x)+\sin (2 (c+d x)))\right )}{7 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \sin \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}\, dx\]
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 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}} \sin {\left (c + d x \right )}\, dx \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 )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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