Optimal. Leaf size=132 \[ \frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3581, 3856,
2720} \begin {gather*} -\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3581
Rule 3856
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{9/2}}{(a+i a \tan (c+d x))^4} \, dx &=\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {\left (5 e^2\right ) \int \frac {(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx}{7 a^2}\\ &=\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\left (5 e^4\right ) \int \sqrt {e \sec (c+d x)} \, dx}{21 a^4}\\ &=\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\left (5 e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^4}\\ &=\frac {10 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 a^4 d}+\frac {4 i e^2 (e \sec (c+d x))^{5/2}}{7 a d (a+i a \tan (c+d x))^3}-\frac {20 i e^4 \sqrt {e \sec (c+d x)}}{21 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 137, normalized size = 1.04 \begin {gather*} \frac {2 e^4 \sec ^4(c+d x) \sqrt {e \sec (c+d x)} \left (5 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-2 i (1+\cos (2 (c+d x))+4 i \sin (2 (c+d x)))\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{21 a^4 d (-i+\tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 228, normalized size = 1.73
method | result | size |
default | \(\frac {2 \left (24 i \left (\cos ^{4}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+24 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-28 i \left (\cos ^{2}\left (d x +c \right )\right )-16 \sin \left (d x +c \right ) \cos \left (d x +c \right )\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \left (\cos ^{4}\left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}{21 a^{4} d \sin \left (d x +c \right )^{4}}\) | \(228\) |
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 = 100, normalized size = 0.76 \begin {gather*} -\frac {2 \, {\left (5 i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c + \frac {9}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (-3 i \, e^{\frac {9}{2}} + 5 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {9}{2}\right )} + 2 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {9}{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}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{21 \, a^{4} d} \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 {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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