Optimal. Leaf size=150 \[ \frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3581, 3854,
3856, 2719} \begin {gather*} \frac {4 i e^2}{13 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{9/2}}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3581
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {1}{(e \sec (c+d x))^{5/2} (a+i a \tan (c+d x))^2} \, dx &=\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (9 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{9/2}} \, dx}{13 a^2}\\ &=\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {7 \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{13 a^2}\\ &=\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {21 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{65 a^2 e^2}\\ &=\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {21 \int \sqrt {\cos (c+d x)} \, dx}{65 a^2 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 e \sin (c+d x)}{13 a^2 d (e \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \sec (c+d x))^{3/2}}+\frac {4 i e^2}{13 d (e \sec (c+d x))^{9/2} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 2.46, size = 149, normalized size = 0.99 \begin {gather*} \frac {(\cos (2 (c+d x))-i \sin (2 (c+d x))) \left (88 i+416 i \cos (2 (c+d x))-8 i \cos (4 (c+d x))-\frac {224 i e^{4 i (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}-356 \sin (2 (c+d x))+18 \sin (4 (c+d x))\right )}{520 a^2 d e^2 \sqrt {e \sec (c+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. 385 vs. \(2 (156 ) = 312\).
time = 1.12, size = 386, normalized size = 2.57
method | result | size |
default | \(-\frac {2 \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \left (-10 i \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )+10 \left (\cos ^{8}\left (d x +c \right )\right )-5 \left (\cos ^{6}\left (d x +c \right )\right )+21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )+21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+2 \left (\cos ^{4}\left (d x +c \right )\right )+14 \left (\cos ^{2}\left (d x +c \right )\right )-21 \cos \left (d x +c \right )\right )}{65 a^{2} d \,e^{5} \sin \left (d x +c \right )^{5}}\) | \(386\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 = 131, normalized size = 0.87 \begin {gather*} \frac {{\left (672 i \, \sqrt {2} e^{\left (7 i \, d x + 7 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \frac {\sqrt {2} {\left (-13 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 373 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 474 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 118 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 35 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\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}}\right )} e^{\left (-7 i \, d x - 7 i \, c - \frac {5}{2}\right )}}{1040 \, a^{2} d} \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*} - \frac {\int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \tan ^{2}{\left (c + d x \right )} - 2 i \left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \tan {\left (c + d x \right )} - \left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx}{a^{2}} \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 {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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