Optimal. Leaf size=191 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{33 a^4 d}+\frac {2 e \sin (c+d x)}{33 a^4 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac {14 i \sqrt {e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac {4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.15, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {4 i e^2}{33 d \left (a^4+i a^4 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}+\frac {2 e \sin (c+d x)}{33 a^4 d \sqrt {e \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{33 a^4 d}+\frac {14 i \sqrt {e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4} \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))^4} \, dx &=\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac {7 \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx}{15 a}\\ &=\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac {14 i \sqrt {e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac {7 \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx}{33 a^2}\\ &=\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac {14 i \sqrt {e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac {4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {e^2 \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{11 a^4}\\ &=\frac {2 e \sin (c+d x)}{33 a^4 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac {14 i \sqrt {e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac {4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\int \sqrt {e \sec (c+d x)} \, dx}{33 a^4}\\ &=\frac {2 e \sin (c+d x)}{33 a^4 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac {14 i \sqrt {e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac {4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{33 a^4}\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{33 a^4 d}+\frac {2 e \sin (c+d x)}{33 a^4 d \sqrt {e \sec (c+d x)}}+\frac {2 i \sqrt {e \sec (c+d x)}}{15 d (a+i a \tan (c+d x))^4}+\frac {14 i \sqrt {e \sec (c+d x)}}{165 a d (a+i a \tan (c+d x))^3}+\frac {4 i e^2}{33 d (e \sec (c+d x))^{3/2} \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 137, normalized size = 0.72 \begin {gather*} \frac {\sec ^4(c+d x) \sqrt {e \sec (c+d x)} \left (40 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (4 (c+d x))+i \sin (4 (c+d x)))+i (64+112 \cos (2 (c+d x))+48 \cos (4 (c+d x))+54 i \sin (2 (c+d x))+37 i \sin (4 (c+d x)))\right )}{660 a^4 d (-i+\tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 224, normalized size = 1.17
method | result | size |
default | \(\frac {2 \sqrt {\frac {e}{\cos \left (d x +c \right )}}\, \left (88 i \left (\cos ^{8}\left (d x +c \right )\right )+88 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )-60 i \left (\cos ^{6}\left (d x +c \right )\right )-16 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+3 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+5 \sin \left (d x +c \right ) \cos \left (d x +c \right )\right )}{165 a^{4} d}\) | \(224\) |
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 = 124, normalized size = 0.65 \begin {gather*} \frac {{\left (-80 i \, \sqrt {2} e^{\left (8 i \, d x + 8 i \, c + \frac {1}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (11 i \, e^{\frac {1}{2}} + 85 i \, e^{\left (8 i \, d x + 8 i \, c + \frac {1}{2}\right )} + 166 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {1}{2}\right )} + 128 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {1}{2}\right )} + 58 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 (-8 i \, d x - 8 i \, c\right )}}{1320 \, a^{4} 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 {\int \frac {\sqrt {e \sec {\left (c + d x \right )}}}{\tan ^{4}{\left (c + d x \right )} - 4 i \tan ^{3}{\left (c + d x \right )} - 6 \tan ^{2}{\left (c + d x \right )} + 4 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{4}} \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 )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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