Optimal. Leaf size=97 \[ \frac {2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 97, 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, 2716,
2721, 2720} \begin {gather*} \frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2720
Rule 2721
Rule 2748
Rubi steps
\begin {align*} \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {2 a}{3 d e (e \cos (c+d x))^{3/2}}+a \int \frac {1}{(e \cos (c+d x))^{5/2}} \, dx\\ &=\frac {2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac {a \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac {2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (a \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 e^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac {2 a \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 86, normalized size = 0.89 \begin {gather*} \frac {2 a \left (\cos (c+d x)-\sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (-1+\sin (c+d x))\right )}{3 d e^2 \sqrt {e \cos (c+d x)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.64, size = 189, normalized size = 1.95
method | result | size |
default | \(-\frac {2 \left (2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d}\) | \(189\) |
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 = 108, normalized size = 1.11 \begin {gather*} \frac {{\left (-i \, \sqrt {2} a \sin \left (d x + c\right ) + i \, \sqrt {2} a\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (i \, \sqrt {2} a \sin \left (d x + c\right ) - i \, \sqrt {2} a\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, a \sqrt {\cos \left (d x + c\right )}}{3 \, {\left (d e^{\frac {5}{2}} \sin \left (d x + c\right ) - d e^{\frac {5}{2}}\right )}} \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 \frac {a+a\,\sin \left (c+d\,x\right )}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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