Optimal. Leaf size=67 \[ -\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{5/6}} \]
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Rubi [A] time = 0.0300104, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2652, 2651} \[ -\frac{2\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{5/6}} \]
Antiderivative was successfully verified.
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Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^{4/3} \, dx &=\frac{\left (a \sqrt [3]{a+a \sin (c+d x)}\right ) \int (1+\sin (c+d x))^{4/3} \, dx}{\sqrt [3]{1+\sin (c+d x)}}\\ &=-\frac{2\ 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)}}{d (1+\sin (c+d x))^{5/6}}\\ \end{align*}
Mathematica [C] time = 1.6485, size = 341, normalized size = 5.09 \[ \frac{(a (\sin (c+d x)+1))^{4/3} \left (-\frac{3}{2} (\cos (c+d x)-5) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{3 (-1)^{3/4} e^{-\frac{3}{2} i (c+d x)} \left (e^{i (c+d x)}+i\right ) \left (-2 \left (1+i e^{-i (c+d x)}\right )^{2/3} \left (1+e^{2 i (c+d x)}\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )\right )+5 i \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-i e^{-i (c+d x)}\right ) \sqrt{2-2 \sin (c+d x)}+20 e^{i (c+d x)} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-i e^{-i (c+d x)}\right ) \sqrt{\cos ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )}\right )}{4 \sqrt{2} \left (1+i e^{-i (c+d x)}\right )^{2/3} \sqrt{i e^{-i (c+d x)} \left (e^{i (c+d x)}-i\right )^2}}\right )}{2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.005, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}\, 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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{4}{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 ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{4}{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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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