Optimal. Leaf size=137 \[ -\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e} \]
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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, 2720} \begin {gather*} \frac {6 a^2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}+\frac {6 a^2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rule 2757
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx &=-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {1}{7} (9 a) \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx\\ &=-\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {1}{7} \left (9 a^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {1}{7} \left (3 a^2 e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac {\left (3 a^2 e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 \sqrt {e \cos (c+d x)}}\\ &=-\frac {18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {6 a^2 e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}+\frac {6 a^2 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac {2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 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 = 0.05, size = 66, normalized size = 0.48 \begin {gather*} -\frac {16 \sqrt [4]{2} a^2 (e \cos (c+d x))^{5/2} \, _2F_1\left (-\frac {9}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.74, size = 203, normalized size = 1.48
method | result | size |
default | \(-\frac {2 a^{2} e^{2} \left (-80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-112 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-20 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-84 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+14 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(203\) |
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 = 118, normalized size = 0.86 \begin {gather*} \frac {-15 i \, \sqrt {2} a^{2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} a^{2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (14 \, a^{2} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 5 \, {\left (a^{2} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} - 3 \, a^{2} e^{\frac {3}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{35 \, 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 {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\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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