Optimal. Leaf size=75 \[ \frac{3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{2/3} d e \sqrt{a \sin (c+d x)+a} \sqrt [3]{e \cos (c+d x)}} \]
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Rubi [A] time = 0.0945724, antiderivative size = 75, 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} \[ \frac{3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{2/3} d e \sqrt{a \sin (c+d x)+a} \sqrt [3]{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{1}{(e \cos (c+d x))^{4/3} \sqrt{a+a \sin (c+d x)}} \, dx &=\frac{\left (a^2 \sqrt [6]{a-a \sin (c+d x)} \sqrt [6]{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-a x)^{7/6} (a+a x)^{5/3}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt [3]{e \cos (c+d x)}}\\ &=\frac{\left (a \sqrt [6]{a-a \sin (c+d x)} \left (\frac{a+a \sin (c+d x)}{a}\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{1}{2}+\frac{x}{2}\right )^{5/3} (a-a x)^{7/6}} \, dx,x,\sin (c+d x)\right )}{2\ 2^{2/3} d e \sqrt [3]{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}\\ &=\frac{3 \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{2/3}}{2^{2/3} d e \sqrt [3]{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0869293, size = 75, normalized size = 1. \[ \frac{3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left (-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{2/3} d e \sqrt{a (\sin (c+d x)+1)} \sqrt [3]{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.11, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{4}{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{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{4}{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{2}{3}} \sqrt{a \sin \left (d x + c\right ) + a}}{a e^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a e^{2} \cos \left (d x + c\right )^{2}}, 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{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{4}{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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