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 = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 2651
Rule 2652
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*}
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Mathematica [C] time = 6.11, 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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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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 \[ \int \frac {1}{\left (a \sin {\left (c + d x \right )} + a\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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