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 = 0.10, 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 = {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 69
Rule 70
Rule 2689
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*}
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Mathematica [A] time = 0.16, 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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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \cos \left (d x + c\right )\right )^{\frac {1}{3}} e^{2} \cos \left (d x + c\right )^{2}}{\sqrt {a \sin \left (d x + c\right ) + a}}, x\right ) \]
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 = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \cos \left (d x +c \right )\right )^{\frac {7}{3}}}{\sqrt {a +a \sin \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{3}}}{\sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/3}}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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