Optimal. Leaf size=137 \[ -\frac {22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}+\frac {22 a^2 e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {22 a^2 e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 (e \cos (c+d x))^{7/2} \left (a^2+a^2 \sin (c+d x)\right )}{9 d e} \]
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Rubi [A]
time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2757, 2748,
2715, 2721, 2719} \begin {gather*} \frac {22 a^2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}-\frac {22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{7/2}}{9 d e}+\frac {22 a^2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 2721
Rule 2748
Rule 2757
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2 \, dx &=-\frac {2 (e \cos (c+d x))^{7/2} \left (a^2+a^2 \sin (c+d x)\right )}{9 d e}+\frac {1}{9} (11 a) \int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x)) \, dx\\ &=-\frac {22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cos (c+d x))^{7/2} \left (a^2+a^2 \sin (c+d x)\right )}{9 d e}+\frac {1}{9} \left (11 a^2\right ) \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}+\frac {22 a^2 e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 (e \cos (c+d x))^{7/2} \left (a^2+a^2 \sin (c+d x)\right )}{9 d e}+\frac {1}{15} \left (11 a^2 e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}+\frac {22 a^2 e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 (e \cos (c+d x))^{7/2} \left (a^2+a^2 \sin (c+d x)\right )}{9 d e}+\frac {\left (11 a^2 e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}+\frac {22 a^2 e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {22 a^2 e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 (e \cos (c+d x))^{7/2} \left (a^2+a^2 \sin (c+d x)\right )}{9 d e}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.07, size = 66, normalized size = 0.48 \begin {gather*} -\frac {16\ 2^{3/4} a^2 (e \cos (c+d x))^{7/2} \, _2F_1\left (-\frac {11}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{7 d e (1+\sin (c+d x))^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.09, size = 260, normalized size = 1.90
method | result | size |
default | \(\frac {2 a^{2} e^{3} \left (-1120 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+2240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1440 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1064 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+2880 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2160 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+84 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+720 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-90 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(260\) |
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 = 0.11, size = 131, normalized size = 0.96 \begin {gather*} \frac {231 i \, \sqrt {2} a^{2} e^{\frac {5}{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 ) - 231 i \, \sqrt {2} a^{2} e^{\frac {5}{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 (90 \, a^{2} \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} + 7 \, {\left (5 \, a^{2} \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} - 11 \, a^{2} \cos \left (d x + c\right ) e^{\frac {5}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{315 \, d} \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]
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 {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\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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