Optimal. Leaf size=130 \[ \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3596, 3567,
3853, 3856, 2719} \begin {gather*} \frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \sin (c+d x) \cos ^3(c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \sin (c+d x) \cos (c+d x)}{5 d (e \cos (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
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))^{7/2}} \, dx &=\frac {\int (e \sec (c+d x))^{7/2} (a+i a \tan (c+d x)) \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {a \int (e \sec (c+d x))^{7/2} \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {\left (3 a e^2\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac {\left (3 a e^4\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}-\frac {\left (3 a \cos ^{\frac {7}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 (e \cos (c+d x))^{7/2}}\\ &=\frac {2 i a}{7 d (e \cos (c+d x))^{7/2}}-\frac {6 a \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d (e \cos (c+d x))^{7/2}}+\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}+\frac {6 a \cos ^3(c+d x) \sin (c+d x)}{5 d (e \cos (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 6.23, size = 596, normalized size = 4.58 \begin {gather*} -\frac {i \cos ^{\frac {9}{2}}(c+d x) \left (-\frac {3 i}{5}+\frac {3 \cot (c)}{5}\right ) \left (-\frac {2 i \sqrt {2} e^{i d x} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )}{3 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}-\frac {2 \left (\cos \left (\frac {c}{2}\right )+i \sin \left (\frac {c}{2}\right )\right )^2 \left (2 i \cos (c+d x)+F\left (\left .\text {ArcSin}\left (\sqrt {-i \cos (c+d x)+\sin (c+d x)}\right )\right |-1\right ) (-i \cos (c+d x)-\sin (c+d x)) \sqrt {1-i \cos (c+d x)+\sin (c+d x)} \sqrt {-i \cos (c+d x)+\cos (2 (c+d x))+\sin (c+d x)+i \sin (2 (c+d x))}+E\left (\left .\text {ArcSin}\left (\sqrt {-i \cos (c+d x)+\sin (c+d x)}\right )\right |-1\right ) \sqrt {1-i \cos (c+d x)+\sin (c+d x)} (i \cos (c+d x)+\sin (c+d x)) \sqrt {-i \cos (c+d x)+\cos (2 (c+d x))+\sin (c+d x)+i \sin (2 (c+d x))}\right )}{\sqrt {\cos (c+d x)}}\right ) (a+i a \tan (c+d x))}{2 d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))}+\frac {\cos ^5(c+d x) \left (\csc (c) \sec (c) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right )+\sec ^4(c+d x) \left (\frac {2}{7} i \cos (c)+\frac {2 \sin (c)}{7}\right )+\sec (c) \sec ^3(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \sin (d x)+\sec (c) \sec (c+d x) \left (\frac {6 \cos (c)}{5}-\frac {6}{5} i \sin (c)\right ) \sin (d x)+\sec ^2(c+d x) \left (\frac {2 \cos (c)}{5}-\frac {2}{5} i \sin (c)\right ) \tan (c)\right ) (a+i a \tan (c+d x))}{d (e \cos (c+d x))^{7/2} (\cos (d x)+i \sin (d x))} \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. 395 vs. \(2 (137 ) = 274\).
time = 3.67, size = 396, normalized size = 3.05
method | result | size |
default | \(\frac {2 \left (336 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+252 \EllipticE \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}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+280 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-126 \EllipticE \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}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\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 )-5 i \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{35 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \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^{3} d}\) | \(396\) |
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 = 221, normalized size = 1.70 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (21 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 77 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 23 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, a e^{\left (2 i \, d x + 2 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 )} + 21 \, {\left (i \, \sqrt {2} a e^{\left (8 i \, d x + 8 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, \sqrt {2} a e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, \sqrt {2} a e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {2} a\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{35 \, {\left (d e^{\frac {7}{2}} + d e^{\left (8 i \, d x + 8 i \, c + \frac {7}{2}\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )}} \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 \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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