Optimal. Leaf size=152 \[ -\frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^2 d}+\frac {14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 152, 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, 3853,
3856, 2719} \begin {gather*} -\frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 e^5 \sin (c+d x) \sqrt {e \sec (c+d x)}}{5 a^2 d}+\frac {14 e^3 \sin (c+d x) (e \sec (c+d x))^{5/2}}{15 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 3581
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{11/2}}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (7 e^2\right ) \int (e \sec (c+d x))^{7/2} \, dx}{3 a^2}\\ &=\frac {14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (7 e^4\right ) \int (e \sec (c+d x))^{3/2} \, dx}{5 a^2}\\ &=\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^2 d}+\frac {14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {\left (7 e^6\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a^2}\\ &=\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^2 d}+\frac {14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac {\left (7 e^6\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=-\frac {14 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 e^5 \sqrt {e \sec (c+d x)} \sin (c+d x)}{5 a^2 d}+\frac {14 e^3 (e \sec (c+d x))^{5/2} \sin (c+d x)}{15 a^2 d}-\frac {4 i e^2 (e \sec (c+d x))^{7/2}}{3 d \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 = 1.17, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 i e^5 e^{i (c+d x)} \left (-47-56 e^{2 i (c+d x)}-21 e^{4 i (c+d x)}+7 \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 )\right ) \sqrt {e \sec (c+d x)}}{15 a^2 d \left (1+e^{2 i (c+d x)}\right )^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. 373 vs. \(2 (158 ) = 316\).
time = 0.70, size = 374, normalized size = 2.46
method | result | size |
default | \(\frac {2 \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (21 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-21 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+21 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-21 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-21 \left (\cos ^{3}\left (d x +c \right )\right )-10 i \cos \left (d x +c \right ) \sin \left (d x +c \right )+24 \left (\cos ^{2}\left (d x +c \right )\right )-3\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {11}{2}} \left (\cos ^{3}\left (d x +c \right )\right )}{15 a^{2} d \sin \left (d x +c \right )^{5}}\) | \(374\) |
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 = 0.10, size = 158, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (21 i \, e^{\left (5 i \, d x + 5 i \, c + \frac {11}{2}\right )} + 56 i \, e^{\left (3 i \, d x + 3 i \, c + \frac {11}{2}\right )} + 47 i \, e^{\left (i \, d x + i \, c + \frac {11}{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}} + 21 \, {\left (i \, \sqrt {2} e^{\frac {11}{2}} + i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c + \frac {11}{2}\right )} + 2 i \, \sqrt {2} e^{\left (2 i \, d x + 2 i \, c + \frac {11}{2}\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )\right )}}{15 \, {\left (a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\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 {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{11/2}}{{\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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