Optimal. Leaf size=105 \[ \frac {2 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 a d}-\frac {2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac {2 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a d} \]
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Rubi [A]
time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3582, 3853,
3856, 2720} \begin {gather*} \frac {2 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)}}{3 a d}+\frac {2 e^3 \sin (c+d x) (e \sec (c+d x))^{3/2}}{3 a d}-\frac {2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3582
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{9/2}}{a+i a \tan (c+d x)} \, dx &=-\frac {2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac {e^2 \int (e \sec (c+d x))^{5/2} \, dx}{a}\\ &=-\frac {2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac {2 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a d}+\frac {e^4 \int \sqrt {e \sec (c+d x)} \, dx}{3 a}\\ &=-\frac {2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac {2 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a d}+\frac {\left (e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a}\\ &=\frac {2 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)}}{3 a d}-\frac {2 i e^2 (e \sec (c+d x))^{5/2}}{5 a d}+\frac {2 e^3 (e \sec (c+d x))^{3/2} \sin (c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 62, normalized size = 0.59 \begin {gather*} \frac {e^2 (e \sec (c+d x))^{5/2} \left (-6 i+10 \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \sin (2 (c+d x))\right )}{15 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 202, normalized size = 1.92
method | result | size |
default | \(\frac {2 \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \left (\cos ^{3}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \left (\cos ^{2}\left (d x +c \right )\right )+5 \sin \left (d x +c \right ) \cos \left (d x +c \right )-3 i\right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \left (\cos ^{2}\left (d x +c \right )\right )}{15 a d \sin \left (d x +c \right )^{4}}\) | \(202\) |
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 = 141, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (\frac {\sqrt {2} {\left (-5 i \, e^{\frac {9}{2}} + 5 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {9}{2}\right )} + 12 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}} + 5 \, {\left (i \, \sqrt {2} e^{\frac {9}{2}} + i \, \sqrt {2} e^{\left (4 i \, d x + 4 i \, c + \frac {9}{2}\right )} + 2 i \, \sqrt {2} e^{\left (2 i \, d x + 2 i \, c + \frac {9}{2}\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{15 \, {\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \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}}{a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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