Optimal. Leaf size=343 \[ \frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {10 a c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt {a \sin (e+f x)+a}}+\frac {2 a c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt {a \sin (e+f x)+a}}-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 1.74, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ \frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {10 a c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt {a \sin (e+f x)+a}}+\frac {2 a c^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt {a \sin (e+f x)+a}}-\frac {2 a (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {a \sin (e+f x)+a}} \]
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))^{7/2} \, dx &=-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{11} (3 a) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{\sqrt {a+a \sin (e+f x)}} \, dx\\ &=\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{11} (5 a c) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx\\ &=\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {1}{7} \left (5 a c^2\right ) \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 {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\left (a c^3\right ) \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^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\left (a c^4\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^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (a c^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}+\frac {\left (a c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {2 a c^4 (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 f g \sqrt {a+a \sin (e+f x)}}+\frac {10 a c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{77 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 8.02, size = 311, normalized size = 0.91 \[ \frac {c^4 g e^{-5 i (e+f x)} \left (e^{i (e+f x)}-i\right ) \left (4928 e^{7 i (e+f x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (e+f x)}\right )+\sqrt {1+e^{2 i (e+f x)}} \left (154 e^{i (e+f x)}+423 i e^{2 i (e+f x)}-308 e^{3 i (e+f x)}+1374 i e^{4 i (e+f x)}-7392 e^{5 i (e+f x)}+1374 i e^{6 i (e+f x)}+308 e^{7 i (e+f x)}+423 i e^{8 i (e+f x)}-154 e^{9 i (e+f x)}-21 i e^{10 i (e+f x)}-21 i\right )\right ) \sqrt {a (\sin (e+f x)+1)} \sqrt {g \cos (e+f x)}}{3696 f \left (e^{i (e+f x)}+i\right ) \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} g \cos \left (f x + e\right ) - {\left (c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} g \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}, x\right ) \]
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 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.76, size = 425, normalized size = 1.24 \[ \frac {2 \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {7}{2}} \left (-21 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+77 \left (\cos ^{6}\left (f x +e \right )\right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+132 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-154 \left (\cos ^{4}\left (f x +e \right )\right )-154 \left (\cos ^{2}\left (f x +e \right )\right )+231 \cos \left (f x +e \right )\right ) \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{231 f \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 \left (\cos ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )+4\right ) \cos \left (f x +e \right )^{3} \sin \left (f x +e \right )} \]
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 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}\,{d x} \]
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 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]
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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