Optimal. Leaf size=152 \[ \frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 e \sin (c+d x)}{117 a^3 d (e \sec (c+d x))^{3/2}}+\frac {2 i}{13 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {28 i e^2}{117 d (e \sec (c+d x))^{5/2} \left (a^3+i a^3 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3583, 3581,
3854, 3856, 2719} \begin {gather*} \frac {28 i e^2}{117 d \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{5/2}}+\frac {14 e \sin (c+d x)}{117 a^3 d (e \sec (c+d x))^{3/2}}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{13 d (a+i a \tan (c+d x))^3 \sqrt {e \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3581
Rule 3583
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3} \, dx &=\frac {2 i}{13 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {7 \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2} \, dx}{13 a}\\ &=\frac {2 i}{13 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {28 i e^2}{117 d (e \sec (c+d x))^{5/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\left (35 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{117 a^3}\\ &=\frac {14 e \sin (c+d x)}{117 a^3 d (e \sec (c+d x))^{3/2}}+\frac {2 i}{13 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {28 i e^2}{117 d (e \sec (c+d x))^{5/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {7 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{39 a^3}\\ &=\frac {14 e \sin (c+d x)}{117 a^3 d (e \sec (c+d x))^{3/2}}+\frac {2 i}{13 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {28 i e^2}{117 d (e \sec (c+d x))^{5/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {7 \int \sqrt {\cos (c+d x)} \, dx}{39 a^3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 e \sin (c+d x)}{117 a^3 d (e \sec (c+d x))^{3/2}}+\frac {2 i}{13 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3}+\frac {28 i e^2}{117 d (e \sec (c+d x))^{5/2} \left (a^3+i a^3 \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 = 1.61, size = 145, normalized size = 0.95 \begin {gather*} \frac {\sqrt {e \sec (c+d x)} (i \cos (3 (c+d x))+\sin (3 (c+d x))) \left (62+176 \cos (2 (c+d x))+114 \cos (4 (c+d x))-56 e^{4 i (c+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 )+126 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))\right )}{468 a^3 d e} \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. 394 vs. \(2 (156 ) = 312\).
time = 1.09, size = 395, normalized size = 2.60
method | result | size |
default | \(-\frac {2 \left (-36 i \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )+36 \left (\cos ^{8}\left (d x +c \right )\right )+13 i \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )-31 \left (\cos ^{6}\left (d x +c \right )\right )+21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )+21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-21 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+2 \left (\cos ^{4}\left (d x +c \right )\right )+14 \left (\cos ^{2}\left (d x +c \right )\right )-21 \cos \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \sqrt {\frac {e}{\cos \left (d x +c \right )}}}{117 a^{3} d \sin \left (d x +c \right )^{5} e}\) | \(395\) |
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 = 120, normalized size = 0.79 \begin {gather*} \frac {{\left (336 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 (219 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 302 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 124 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 50 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 9 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 {1}{2}\right )}}{936 \, a^{3} 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 {i \int \frac {1}{\sqrt {e \sec {\left (c + d x \right )}} \tan ^{3}{\left (c + d x \right )} - 3 i \sqrt {e \sec {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )} - 3 \sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )} + i \sqrt {e \sec {\left (c + d x \right )}}}\, dx}{a^{3}} \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}{\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,{\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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