Optimal. Leaf size=140 \[ -\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}+\frac {22 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e} \]
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Rubi [A]
time = 0.11, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2757, 2748,
2721, 2719} \begin {gather*} -\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {22 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{3/2}}{35 d e}+\frac {22 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rule 2748
Rule 2757
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}+\frac {1}{7} (11 a) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e}+\frac {1}{5} \left (11 a^2\right ) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x)) \, dx\\ &=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e}+\frac {1}{5} \left (11 a^3\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e}+\frac {\left (11 a^3 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}+\frac {22 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 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 = 0.03, size = 66, normalized size = 0.47 \begin {gather*} -\frac {16\ 2^{3/4} a^3 (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac {11}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (1+\sin (c+d x))^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.88, size = 214, normalized size = 1.53
method | result | size |
default | \(\frac {2 a^{3} e \left (240 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-480 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-200 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-126 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+440 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-125 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(214\) |
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 = 126, normalized size = 0.90 \begin {gather*} \frac {231 i \, \sqrt {2} a^{3} e^{\frac {1}{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^{3} e^{\frac {1}{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 (15 \, a^{3} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 63 \, a^{3} \cos \left (d x + c\right ) e^{\frac {1}{2}} \sin \left (d x + c\right ) - 140 \, a^{3} \cos \left (d x + c\right ) e^{\frac {1}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{105 \, 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 \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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