Optimal. Leaf size=66 \[ -\frac {\cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [6]{1+\sin (c+d x)}}{\sqrt [6]{2} d (a+a \sin (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.02, 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 = {2731, 2730}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2730
Rule 2731
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] Result contains complex when optimal does not.
time = 13.48, size = 604, normalized size = 9.15 \begin {gather*} \frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-3+\frac {3 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{d (a (1+\sin (c+d x)))^{2/3}}-\frac {2 \sqrt {2} \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [6]{1+\sin (c+d x)} \left (-\frac {i \sqrt [3]{\cos \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )} \left (-\frac {3 i \left (e^{-i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}+e^{i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}\right )^{2/3} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-e^{2 i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}\right )}{2^{2/3} \left (1+e^{2 i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}\right )^{2/3}}-\frac {3 i e^{i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )} \sqrt [3]{1+e^{2 i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-e^{2 i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}\right )}{2\ 2^{2/3} \sqrt [3]{e^{-i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}+e^{i \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}}}\right )}{2 \sqrt [6]{1+\cos \left (2 \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )\right )}}+\frac {3 \cos ^2\left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2\left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )\right ) \sin \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}{5 \sqrt [6]{1+\cos \left (2 \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )\right )} \sqrt {\sin ^2\left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right )}}\right )}{d (a (1+\sin (c+d x)))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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 \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 \frac {1}{\left (a \sin {\left (c + d x \right )} + a\right )^{\frac {2}{3}}}\, 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 = -0.02 \begin {gather*} \int \frac {1}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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