Optimal. Leaf size=96 \[ \frac {2 i a}{5 d (e \cos (c+d x))^{5/2}}+\frac {2 a \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d (e \cos (c+d x))^{5/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{3 d (e \cos (c+d x))^{5/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3567,
3853, 3856, 2720} \begin {gather*} \frac {2 i a}{5 d (e \cos (c+d x))^{5/2}}+\frac {2 a \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d (e \cos (c+d x))^{5/2}}+\frac {2 a \sin (c+d x) \cos (c+d x)}{3 d (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3567
Rule 3596
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \frac {a+i a \tan (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {\int (e \sec (c+d x))^{5/2} (a+i a \tan (c+d x)) \, dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=\frac {2 i a}{5 d (e \cos (c+d x))^{5/2}}+\frac {a \int (e \sec (c+d x))^{5/2} \, dx}{(e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=\frac {2 i a}{5 d (e \cos (c+d x))^{5/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{3 d (e \cos (c+d x))^{5/2}}+\frac {\left (a e^2\right ) \int \sqrt {e \sec (c+d x)} \, dx}{3 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}}\\ &=\frac {2 i a}{5 d (e \cos (c+d x))^{5/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{3 d (e \cos (c+d x))^{5/2}}+\frac {\left (a \cos ^{\frac {5}{2}}(c+d x)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 (e \cos (c+d x))^{5/2}}\\ &=\frac {2 i a}{5 d (e \cos (c+d x))^{5/2}}+\frac {2 a \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d (e \cos (c+d x))^{5/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{3 d (e \cos (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 57, normalized size = 0.59 \begin {gather*} \frac {a \left (6 i+10 \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \sin (2 (c+d x))\right )}{15 d (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 282 vs. \(2 (107 ) = 214\).
time = 2.14, size = 283, normalized size = 2.95
method | result | size |
default | \(-\frac {2 \left (20 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-20 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{15 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d}\) | \(283\) |
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 = 0.08, size = 178, normalized size = 1.85 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (5 i \, a e^{\left (5 i \, d x + 5 i \, c\right )} - 12 i \, a e^{\left (3 i \, d x + 3 i \, c\right )} - 5 i \, a e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + 5 \, {\left (i \, \sqrt {2} a e^{\left (6 i \, d x + 6 i \, c\right )} + 3 i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{15 \, {\left (d e^{\frac {5}{2}} + d e^{\left (6 i \, d x + 6 i \, c + \frac {5}{2}\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {5}{2}\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\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\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\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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