Optimal. Leaf size=292 \[ \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {2 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{65 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A]
time = 0.91, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2931,
2921, 2721, 2719} \begin {gather*} \frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{65 c^4 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rule 2921
Rule 2929
Rule 2931
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx &=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {(3 a) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{13 c}\\ &=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {a \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}{13 c^2}\\ &=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {a \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}{65 c^3}\\ &=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{65 c^4}\\ &=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {(a g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{65 c^4 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{65 c^4 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {2 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{65 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 1.94, size = 291, normalized size = 1.00 \begin {gather*} \frac {4 e^{3 i (e+f x)} \left (e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right ) g\right )^{3/2} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-1+149 i e^{i (e+f x)}+44 e^{2 i (e+f x)}-64 i e^{3 i (e+f x)}+21 e^{4 i (e+f x)}+3 i e^{5 i (e+f x)}\right )-i \left (-i+e^{i (e+f x)}\right )^7 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{195 c^4 \left (1-i e^{i (e+f x)}\right ) \left (-i+e^{i (e+f x)}\right )^6 \sqrt {i c e^{-i (e+f x)} \left (-i+e^{i (e+f x)}\right )^2} \left (1+e^{2 i (e+f x)}\right )^{3/2} f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.26, size = 1126, normalized size = 3.86
method | result | size |
default | \(\text {Expression too large to display}\) | \(1126\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.14, size = 350, normalized size = 1.20 \begin {gather*} \frac {2 \, {\left (12 \, g \cos \left (f x + e\right )^{2} - {\left (3 \, g \cos \left (f x + e\right )^{2} - 23 \, g\right )} \sin \left (f x + e\right ) + 7 \, g\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} - 3 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{4} - 8 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 4 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) + 8 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 3 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{4} + 8 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 4 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) - 8 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{195 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {{\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 )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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