Optimal. Leaf size=235 \[ \frac {2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {6 a c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 1.13, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ \frac {2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {6 a c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2842
Rule 2851
Rubi steps
\begin {align*} \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx &=-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{7} (3 a) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx\\ &=\frac {6 a c (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{5} (3 a c) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx\\ &=\frac {2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{5} \left (3 a c^2\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (3 a c^2 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (3 a c^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 1.72, size = 255, normalized size = 1.09 \[ \frac {c^2 g e^{-3 i (e+f x)} \left (e^{i (e+f x)}-i\right ) \left (112 e^{5 i (e+f x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (e+f x)}\right )+\sqrt {1+e^{2 i (e+f x)}} \left (-14 e^{i (e+f x)}+15 i e^{2 i (e+f x)}-168 e^{3 i (e+f x)}+15 i e^{4 i (e+f x)}+14 e^{5 i (e+f x)}+5 i e^{6 i (e+f x)}+5 i\right )\right ) \sqrt {a (\sin (e+f x)+1)} \sqrt {g \cos (e+f x)}}{140 f \left (e^{i (e+f x)}+i\right ) \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (c g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - c g \cos \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.62, size = 372, normalized size = 1.58 \[ -\frac {2 \left (21 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-21 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+5 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+21 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-21 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-7 \left (\cos ^{4}\left (f x +e \right )\right )-14 \left (\cos ^{2}\left (f x +e \right )\right )+21 \cos \left (f x +e \right )\right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{35 f \left (\sin \left (f x +e \right )-1\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )^{3}} \]
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 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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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