Optimal. Leaf size=114 \[ \frac {14 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt {\cos (c+d x)}}+\frac {14 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^2 d}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2+a^2 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2759, 2715,
2721, 2719} \begin {gather*} \frac {14 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a^2 d \sqrt {\cos (c+d x)}}+\frac {14 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^2 d}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 2721
Rule 2759
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx &=\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (7 e^2\right ) \int (e \cos (c+d x))^{5/2} \, dx}{3 a^2}\\ &=\frac {14 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^2 d}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (7 e^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 a^2}\\ &=\frac {14 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^2 d}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (7 e^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^2 \sqrt {\cos (c+d x)}}\\ &=\frac {14 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt {\cos (c+d x)}}+\frac {14 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^2 d}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.09, size = 66, normalized size = 0.58 \begin {gather*} -\frac {2\ 2^{3/4} (e \cos (c+d x))^{11/2} \, _2F_1\left (\frac {1}{4},\frac {11}{4};\frac {15}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{11 a^2 d e (1+\sin (c+d x))^{11/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.82, size = 190, normalized size = 1.67
method | result | size |
default | \(\frac {2 e^{5} \left (-24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+40 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-40 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(190\) |
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 = 0.10, size = 102, normalized size = 0.89 \begin {gather*} \frac {21 i \, \sqrt {2} e^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} e^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, \cos \left (d x + c\right ) e^{\frac {9}{2}} \sin \left (d x + c\right ) - 10 \, \cos \left (d x + c\right ) e^{\frac {9}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{15 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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