Optimal. Leaf size=175 \[ -\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3579, 3567,
3853, 3856, 2719} \begin {gather*} -\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 d}+\frac {22 i \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{35 d}+\frac {2 i a (a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2}}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3567
Rule 3579
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^3 \, dx &=\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {1}{7} (11 a) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2 \, dx\\ &=\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac {1}{5} \left (11 a^2\right ) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx\\ &=\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}+\frac {1}{5} \left (11 a^3\right ) \int (e \sec (c+d x))^{3/2} \, dx\\ &=\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}-\frac {1}{5} \left (11 a^3 e^2\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx\\ &=\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}-\frac {\left (11 a^3 e^2\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=-\frac {22 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {22 i a^3 (e \sec (c+d x))^{3/2}}{15 d}+\frac {22 a^3 e \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 i a (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2}{7 d}+\frac {22 i (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}{35 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 2.31, size = 129, normalized size = 0.74 \begin {gather*} \frac {a^3 (e \sec (c+d x))^{3/2} (1+i \tan (c+d x)) \left (-116 i-308 i \cos (2 (c+d x))+77 i e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+77 \sec (c+d x) \sin (3 (c+d x))+17 \tan (c+d x)\right )}{210 d} \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. 391 vs. \(2 (174 ) = 348\).
time = 0.54, size = 392, normalized size = 2.24
method | result | size |
default | \(-\frac {2 a^{3} \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (231 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-231 i \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+231 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-231 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-140 i \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+231 \left (\cos ^{4}\left (d x +c \right )\right )-294 \left (\cos ^{3}\left (d x +c \right )\right )+15 i \sin \left (d x +c \right )+63 \cos \left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}{105 d \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}\) | \(392\) |
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 = 0.10, size = 211, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (231 i \, a^{3} e^{\left (7 i \, d x + 7 i \, c + \frac {3}{2}\right )} + 287 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c + \frac {3}{2}\right )} + 253 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c + \frac {3}{2}\right )} + 77 i \, a^{3} e^{\left (i \, d x + i \, c + \frac {3}{2}\right )}\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{\sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} + 231 \, {\left (i \, \sqrt {2} a^{3} e^{\frac {3}{2}} + i \, \sqrt {2} a^{3} e^{\left (6 i \, d x + 6 i \, c + \frac {3}{2}\right )} + 3 i \, \sqrt {2} a^{3} e^{\left (4 i \, d x + 4 i \, c + \frac {3}{2}\right )} + 3 i \, \sqrt {2} a^{3} e^{\left (2 i \, d x + 2 i \, c + \frac {3}{2}\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i a^{3} \left (\int i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (- 3 \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan {\left (c + d x \right )}\right )\, dx + \int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (c + d x \right )}\, dx + \int \left (- 3 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (c + d x \right )}\right )\, dx\right ) \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 (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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