Optimal. Leaf size=152 \[ \frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{77 a^3 d}+\frac {10 e \sin (c+d x)}{77 a^3 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3}+\frac {20 i e^2}{77 d (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )} \]
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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, 2720} \begin {gather*} \frac {20 i e^2}{77 d \left (a^3+i a^3 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}+\frac {10 e \sin (c+d x)}{77 a^3 d \sqrt {e \sec (c+d x)}}+\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{77 a^3 d}+\frac {2 i \sqrt {e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3581
Rule 3583
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx &=\frac {2 i \sqrt {e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3}+\frac {5 \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx}{11 a}\\ &=\frac {2 i \sqrt {e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3}+\frac {20 i e^2}{77 d (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\left (15 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{77 a^3}\\ &=\frac {10 e \sin (c+d x)}{77 a^3 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3}+\frac {20 i e^2}{77 d (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {5 \int \sqrt {e \sec (c+d x)} \, dx}{77 a^3}\\ &=\frac {10 e \sin (c+d x)}{77 a^3 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3}+\frac {20 i e^2}{77 d (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\left (5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 a^3}\\ &=\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{77 a^3 d}+\frac {10 e \sin (c+d x)}{77 a^3 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{11 d (a+i a \tan (c+d x))^3}+\frac {20 i e^2}{77 d (e \sec (c+d x))^{3/2} \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.54, size = 129, normalized size = 0.85 \begin {gather*} \frac {i \sec ^3(c+d x) \sqrt {e \sec (c+d x)} \left (46 i \cos (c+d x)+22 i \cos (3 (c+d x))-15 \sin (c+d x)+20 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x)))-15 \sin (3 (c+d x))\right )}{154 a^3 d (-i+\tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.12, size = 208, normalized size = 1.37
method | result | size |
default | \(\frac {2 \sqrt {\frac {e}{\cos \left (d x +c \right )}}\, \left (28 i \left (\cos ^{6}\left (d x +c \right )\right )+28 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )-11 i \left (\cos ^{4}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+3 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+5 \sin \left (d x +c \right ) \cos \left (d x +c \right )\right )}{77 a^{3} d}\) | \(208\) |
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 = 112, normalized size = 0.74 \begin {gather*} \frac {{\left (-40 i \, \sqrt {2} e^{\left (6 i \, d x + 6 i \, c + \frac {1}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (7 i \, e^{\frac {1}{2}} + 37 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {1}{2}\right )} + 61 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 31 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {1}{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 (-6 i \, d x - 6 i \, c\right )}}{308 \, 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 {\sqrt {e \sec {\left (c + d x \right )}}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, 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 {\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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