Optimal. Leaf size=77 \[ -\frac {3 \sqrt [6]{\sin (c+d x)+1} (e \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {7}{6};\frac {4}{3};\frac {1}{2} (1-\sin (c+d x))\right )}{2 \sqrt [6]{2} d e \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.09, antiderivative size = 77, 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} \[ -\frac {3 \sqrt [6]{\sin (c+d x)+1} (e \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {7}{6};\frac {4}{3};\frac {1}{2} (1-\sin (c+d x))\right )}{2 \sqrt [6]{2} d e \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 2689
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\left (a^2 (e \cos (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-a x)^{2/3} (a+a x)^{7/6}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt [3]{a-a \sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ &=\frac {\left (a (e \cos (c+d x))^{2/3} \sqrt [6]{\frac {a+a \sin (c+d x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {x}{2}\right )^{7/6} (a-a x)^{2/3}} \, dx,x,\sin (c+d x)\right )}{2 \sqrt [6]{2} d e \sqrt [3]{a-a \sin (c+d x)} \sqrt {a+a \sin (c+d x)}}\\ &=-\frac {3 (e \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {7}{6};\frac {4}{3};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [6]{1+\sin (c+d x)}}{2 \sqrt [6]{2} d e \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 1.00 \[ -\frac {3 \sqrt [6]{\sin (c+d x)+1} (e \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {7}{6};\frac {4}{3};\frac {1}{2} (1-\sin (c+d x))\right )}{2 \sqrt [6]{2} d e \sqrt {a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sqrt {a \sin \left (d x + c\right ) + a}}{a e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )}, x\right ) \]
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 = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \cos \left (d x +c \right )\right )^{\frac {1}{3}} \sqrt {a +a \sin \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {1}{3}} \sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{1/3}\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sqrt [3]{e \cos {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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