Optimal. Leaf size=37 \[ \frac {x}{4}-\frac {i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \]
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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 = {2657} \[ \frac {x}{4}-\frac {i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2657
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] time = 0.03, size = 171, normalized size = 4.62 \[ -\frac {\log (5 \cosh (c+d x)-4 \sinh (c+d x))}{8 d}+\frac {\log (4 \sinh (c+d x)+5 \cosh (c+d x))}{8 d}-\frac {i \tan ^{-1}\left (\frac {2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-2 \sinh \left (\frac {1}{2} (c+d x)\right )}\right )}{4 d}+\frac {i \tan ^{-1}\left (\frac {2 \sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right )}{\sinh \left (\frac {1}{2} (c+d x)\right )+2 \cosh \left (\frac {1}{2} (c+d x)\right )}\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 26, normalized size = 0.70 \[ \frac {\log \left (e^{\left (d x + c\right )} - \frac {1}{3} i\right ) - \log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 28, normalized size = 0.76 \[ \frac {\log \left (3 \, e^{\left (d x + c\right )} - i\right ) - \log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 1.19 \[ -\frac {\ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{4 d}+\frac {\ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 36, normalized size = 0.97 \[ \frac {\log \left (-\frac {6 \, {\left (-i \, e^{\left (-d x - c\right )} + 3\right )}}{6 i \, e^{\left (-d x - c\right )} - 2}\right )}{4 \, d} \]
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 \[ -\frac {\ln \left (-\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{2}+\frac {3}{2}{}\mathrm {i}\right )-\ln \left (\frac {9\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{2}-\frac {3}{2}{}\mathrm {i}\right )}{4\,d} \]
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 \[ \text {Exception raised: NotInvertible} \]
Verification of antiderivative is not currently implemented for this CAS.
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