Optimal. Leaf size=69 \[ -\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.0336768, antiderivative size = 69, 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} \[ -\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (c+d x))^{4/3}} \, dx &=\frac{\sqrt [3]{1+\sin (c+d x)} \int \frac{1}{(1+\sin (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.286556, size = 130, normalized size = 1.88 \[ -\frac{3 \left (\sqrt{2-2 \sin (c+d x)}-2 (\sin (c+d x)+1) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{5 d \sqrt{2-2 \sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, 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 \frac{1}{{\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 (-\frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{2}{3}}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin{\left (c + d x \right )} + a\right )^{\frac{4}{3}}}\, dx \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 \frac{1}{{\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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