Optimal. Leaf size=80 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{3 a d}+\frac {2 i \sqrt {e \sec (c+d x)}}{3 d (a+i a \tan (c+d x))} \]
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Rubi [A]
time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3583, 3856,
2720} \begin {gather*} \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)}}{3 a d}+\frac {2 i \sqrt {e \sec (c+d x)}}{3 d (a+i a \tan (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3583
Rule 3856
Rubi steps
\begin {align*} \int \frac {\sqrt {e \sec (c+d x)}}{a+i a \tan (c+d x)} \, dx &=\frac {2 i \sqrt {e \sec (c+d x)}}{3 d (a+i a \tan (c+d x))}+\frac {\int \sqrt {e \sec (c+d x)} \, dx}{3 a}\\ &=\frac {2 i \sqrt {e \sec (c+d x)}}{3 d (a+i a \tan (c+d x))}+\frac {\left (\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 \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 \sqrt {e \sec (c+d x)}}{3 d (a+i a \tan (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 83, normalized size = 1.04 \begin {gather*} \frac {2 (e \sec (c+d x))^{3/2} \left (\cos (c+d x)+\sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (-i \cos (c+d x)+\sin (c+d x))\right )}{3 a d e (-i+\tan (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 164, normalized size = 2.05
method | result | size |
default | \(\frac {2 \sqrt {\frac {e}{\cos \left (d x +c \right )}}\, \left (i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+i \left (\cos ^{2}\left (d x +c \right )\right )+\sin \left (d x +c \right ) \cos \left (d x +c \right )\right )}{3 a d}\) | \(164\) |
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 = 0.09, size = 88, normalized size = 1.10 \begin {gather*} \frac {{\left (-2 i \, \sqrt {2} e^{\left (2 i \, d x + 2 i \, c + \frac {1}{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \frac {\sqrt {2} {\left (i \, e^{\frac {1}{2}} + 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 (-2 i \, d x - 2 i \, c\right )}}{3 \, a 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 {\left (c + d x \right )} - i}\, dx}{a} \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 )}}}{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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