Optimal. Leaf size=409 \[ -\frac {14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^4 c^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{13 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c^2 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g} \]
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Rubi [A] time = 2.04, antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac {14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c^2 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 a^4 c^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{13 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g} \]
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} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {1}{13} (7 c) \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {1}{143} \left (21 c^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {1}{143} \left (35 a c^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {1}{13} \left (5 a^2 c^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {1}{13} \left (7 a^3 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 {14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {1}{13} \left (7 a^4 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 {14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {\left (7 a^4 c^2 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{13 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}+\frac {\left (7 a^4 c^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{13 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^4 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 )}{13 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g}\\ \end {align*}
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Mathematica [A] time = 2.92, size = 212, normalized size = 0.52 \[ \frac {a^3 c (\sin (e+f x)-1) (\sin (e+f x)+1)^3 \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (\sqrt {\cos (e+f x)} (-1507 \sin (2 (e+f x))-88 \sin (4 (e+f x))+33 \sin (6 (e+f x))+1560 \cos (e+f x)+780 \cos (3 (e+f x))+156 \cos (5 (e+f x)))-7392 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right )}{6864 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{3} c g \cos \left (f x + e\right )^{5} - 2 \, a^{3} c g \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - 2 \, a^{3} c g \cos \left (f x + e\right )^{3}\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}, x\right ) \]
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 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.73, size = 404, normalized size = 0.99 \[ -\frac {2 \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \left (-33 \left (\cos ^{8}\left (f x +e \right )\right )+78 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+88 \left (\cos ^{6}\left (f x +e \right )\right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+22 \left (\cos ^{4}\left (f x +e \right )\right )+154 \left (\cos ^{2}\left (f x +e \right )\right )-231 \cos \left (f x +e \right )\right ) \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}}{429 f \left (2 \sin \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )+2\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )^{5}} \]
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 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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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