Optimal. Leaf size=78 \[ -\frac {3 \sqrt [3]{2} a (e \cos (c+d x))^{5/3} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{2/3}}{5 d e (a+a \sin (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2768, 72, 71}
\begin {gather*} -\frac {3 \sqrt [3]{2} a (\sin (c+d x)+1)^{2/3} (e \cos (c+d x))^{5/3} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (a \sin (c+d x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 2768
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{2/3}}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [6]{a-a x} (a+a x)^{2/3}} \, dx,x,\sin (c+d x)\right )}{d e (a-a \sin (c+d x))^{5/6} (a+a \sin (c+d x))^{5/6}}\\ &=\frac {\left (a^2 (e \cos (c+d x))^{5/3} \left (\frac {a+a \sin (c+d x)}{a}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {x}{2}\right )^{2/3} \sqrt [6]{a-a x}} \, dx,x,\sin (c+d x)\right )}{2^{2/3} d e (a-a \sin (c+d x))^{5/6} (a+a \sin (c+d x))^{3/2}}\\ &=-\frac {3 \sqrt [3]{2} a (e \cos (c+d x))^{5/3} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{2/3}}{5 d e (a+a \sin (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 77, normalized size = 0.99 \begin {gather*} -\frac {3 \sqrt [3]{2} (e \cos (c+d x))^{5/3} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e \sqrt [3]{1+\sin (c+d x)} \sqrt {a (1+\sin (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (e \cos \left (d x +c \right )\right )^{\frac {2}{3}}}{\sqrt {a +a \sin \left (d x +c \right )}}\, dx\]
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 {\left (e \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{2/3}}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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