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 = 0.09, antiderivative size = 75, 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 = {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 69
Rule 70
Rule 2689
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*}
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Mathematica [A] time = 0.09, size = 75, normalized size = 1.00 \[ \frac {3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {5}{3};\frac {5}{6};\frac {1}{2} (1-\sin (c+d x))\right )}{2^{2/3} d e \sqrt {a (\sin (c+d x)+1)} \sqrt [3]{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sqrt {a \sin \left (d x + c\right ) + a}}{a e^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \cos \left (d x +c \right )\right )^{\frac {4}{3}} \sqrt {a +a \sin \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {4}{3}} \sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{4/3}\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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