Optimal. Leaf size=237 \[ -\frac {6 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 a c^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}} \]
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Rubi [A] time = 1.16, antiderivative size = 237, 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 = {2852, 2842, 2640, 2639} \[ -\frac {6 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 a c^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2842
Rule 2852
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}} \, dx &=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}+\frac {3 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{a}\\ &=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {3 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{5 a c}\\ &=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {3 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{5 a c^2}\\ &=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {(3 g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{5 a c^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 a c^2 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {6 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 a c^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.11, size = 134, normalized size = 0.57 \[ -\frac {\sqrt {\cos (e+f x)} (g \cos (e+f x))^{3/2} \left (\sqrt {\cos (e+f x)} (-6 \sin (e+f x)-3 \cos (2 (e+f x))+1)+E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) (6 \cos (e+f x)-3 \sin (2 (e+f x)))\right )}{5 c^2 f (\sin (e+f x)-1)^2 (a (\sin (e+f x)+1))^{3/2} \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g}{a^{2} c^{3} \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - a^{2} c^{3} \cos \left (f x + e\right )^{3}}, x\right ) \]
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 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 877, normalized size = 3.70 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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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