Optimal. Leaf size=95 \[ -\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e}+\frac {6 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2748, 2715,
2721, 2719} \begin {gather*} \frac {6 a e^2 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 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 2721
Rule 2748
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \, dx &=-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e}+a \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} \left (3 a e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\left (3 a e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e}+\frac {6 a e^2 \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 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 79, normalized size = 0.83 \begin {gather*} \frac {(e \cos (c+d x))^{5/2} \left (42 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) (-5 b-5 b \cos (2 (c+d x))+14 a \sin (c+d x))\right )}{35 d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs.
\(2(107)=214\).
time = 4.20, size = 222, normalized size = 2.34
method | result | size |
default | \(\frac {2 e^{3} \left (-80 b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +14 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(222\) |
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 = 105, normalized size = 1.11 \begin {gather*} \frac {21 i \, \sqrt {2} a 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 ) - 21 i \, \sqrt {2} a 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 (5 \, b \cos \left (d x + c\right )^{3} e^{\frac {5}{2}} - 7 \, a \cos \left (d x + c\right ) e^{\frac {5}{2}} \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 )}^{5/2}\,\left (a+b\,\sin \left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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