Optimal. Leaf size=78 \[ -\frac{3 \sqrt [6]{2} a (e \cos (c+d x))^{10/3} \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right )}{5 d e \sqrt [6]{\sin (c+d x)+1} (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.0954889, antiderivative size = 78, normalized size of antiderivative = 1., 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 = {2689, 70, 69} \[ -\frac{3 \sqrt [6]{2} a (e \cos (c+d x))^{10/3} \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right )}{5 d e \sqrt [6]{\sin (c+d x)+1} (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{7/3}}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{\left (a^2 (e \cos (c+d x))^{10/3}\right ) \operatorname{Subst}\left (\int (a-a x)^{2/3} \sqrt [6]{a+a x} \, dx,x,\sin (c+d x)\right )}{d e (a-a \sin (c+d x))^{5/3} (a+a \sin (c+d x))^{5/3}}\\ &=\frac{\left (\sqrt [6]{2} a^2 (e \cos (c+d x))^{10/3}\right ) \operatorname{Subst}\left (\int \sqrt [6]{\frac{1}{2}+\frac{x}{2}} (a-a x)^{2/3} \, dx,x,\sin (c+d x)\right )}{d e (a-a \sin (c+d x))^{5/3} (a+a \sin (c+d x))^{3/2} \sqrt [6]{\frac{a+a \sin (c+d x)}{a}}}\\ &=-\frac{3 \sqrt [6]{2} a (e \cos (c+d x))^{10/3} \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right )}{5 d e \sqrt [6]{1+\sin (c+d x)} (a+a \sin (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.153058, size = 77, normalized size = 0.99 \[ -\frac{3 \sqrt [6]{2} (e \cos (c+d x))^{10/3} \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right )}{5 d e (\sin (c+d x)+1)^{7/6} \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{7}{3}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}}\, 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{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{3}}}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{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 (e \cos \left (d x + c\right )\right )^{\frac{1}{3}} e^{2} \cos \left (d x + c\right )^{2}}{\sqrt{a \sin \left (d x + c\right ) + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{3}}}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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