Optimal. Leaf size=37 \[ \frac {x}{4}-\frac {i \text {ArcTan}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2736}
\begin {gather*} \frac {x}{4}-\frac {i \text {ArcTan}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2736
Rubi steps
\begin {align*} \int \frac {1}{5+3 i \sinh (c+d x)} \, dx &=\frac {x}{4}-\frac {i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(171\) vs. \(2(37)=74\).
time = 0.03, size = 171, normalized size = 4.62 \begin {gather*} -\frac {i \text {ArcTan}\left (\frac {2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-2 \sinh \left (\frac {1}{2} (c+d x)\right )}\right )}{4 d}+\frac {i \text {ArcTan}\left (\frac {\cosh \left (\frac {1}{2} (c+d x)\right )+2 \sinh \left (\frac {1}{2} (c+d x)\right )}{2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )}\right )}{4 d}-\frac {\log (5 \cosh (c+d x)-4 \sinh (c+d x))}{8 d}+\frac {\log (5 \cosh (c+d x)+4 \sinh (c+d x))}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 42, normalized size = 1.14
method | result | size |
risch | \(-\frac {\ln \left (-3 i+{\mathrm e}^{d x +c}\right )}{4 d}+\frac {\ln \left ({\mathrm e}^{d x +c}-\frac {i}{3}\right )}{4 d}\) | \(32\) |
derivativedivides | \(\frac {-\frac {\ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{4}+\frac {\ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{4}}{d}\) | \(42\) |
default | \(\frac {-\frac {\ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{4}+\frac {\ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{4}}{d}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 36, normalized size = 0.97 \begin {gather*} \frac {\log \left (-\frac {6 \, {\left (-i \, e^{\left (-d x - c\right )} + 3\right )}}{6 i \, e^{\left (-d x - c\right )} - 2}\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 0.70 \begin {gather*} \frac {\log \left (e^{\left (d x + c\right )} - \frac {1}{3} i\right ) - \log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 31, normalized size = 0.84 \begin {gather*} \frac {- \frac {\log {\left (e^{d x} - 3 i e^{- c} \right )}}{4} + \frac {\log {\left (e^{d x} - \frac {i e^{- c}}{3} \right )}}{4}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 28, normalized size = 0.76 \begin {gather*} \frac {\log \left (3 \, e^{\left (d x + c\right )} - i\right ) - \log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 32, normalized size = 0.86 \begin {gather*} -\frac {\ln \left (-\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{2}+\frac {3}{2}{}\mathrm {i}\right )-\ln \left (\frac {9\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{2}-\frac {3}{2}{}\mathrm {i}\right )}{4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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