Optimal. Leaf size=66 \[ -\frac{\sqrt [6]{\sin (c+d x)+1} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{\sqrt [6]{2} d (a \sin (c+d x)+a)^{2/3}} \]
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Rubi [A] time = 0.0332749, antiderivative size = 66, 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} \[ -\frac{\sqrt [6]{\sin (c+d x)+1} \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{\sqrt [6]{2} d (a \sin (c+d x)+a)^{2/3}} \]
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))^{2/3}} \, dx &=\frac{(1+\sin (c+d x))^{2/3} \int \frac{1}{(1+\sin (c+d x))^{2/3}} \, dx}{(a+a \sin (c+d x))^{2/3}}\\ &=-\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right ) \sqrt [6]{1+\sin (c+d x)}}{\sqrt [6]{2} d (a+a \sin (c+d x))^{2/3}}\\ \end{align*}
Mathematica [C] time = 6.09823, size = 604, normalized size = 9.15 \[ \frac{2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (\frac{3 \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-3\right )}{d (a (\sin (c+d x)+1))^{2/3}}-\frac{2 \sqrt{2} \sqrt [6]{\sin (c+d x)+1} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (\frac{3 \sin \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right ) \cos ^2\left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2\left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )\right )}{5 \sqrt{\sin ^2\left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )} \sqrt [6]{\cos \left (2 \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )\right )+1}}-\frac{i \left (-\frac{3 i \left (e^{-i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}+e^{i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}\right )^{2/3} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}\right )}{2^{2/3} \left (1+e^{2 i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}\right )^{2/3}}-\frac{3 i e^{i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )} \sqrt [3]{1+e^{2 i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{e^{-i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}+e^{i \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}}}\right ) \sqrt [3]{\cos \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )}}{2 \sqrt [6]{\cos \left (2 \left (\frac{1}{2} (-c-d x)+\frac{\pi }{4}\right )\right )+1}}\right )}{d (a (\sin (c+d x)+1))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( dx+c \right ) \right ) ^{-{\frac{2}{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{2}{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{1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{2}{3}}}, 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{2}{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{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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