Optimal. Leaf size=12 \[ \log (a+\text {eps} \cosh (x)+i x) \]
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Rubi [A] time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6684} \[ \log (a+\text {eps} \cosh (x)+i x) \]
Antiderivative was successfully verified.
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Rule 6684
Rubi steps
\begin {align*} \int \frac {-1+i \text {eps} \sinh (x)}{i a-x+i \text {eps} \cosh (x)} \, dx &=\log (a+i x+\text {eps} \cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.09, size = 12, normalized size = 1.00 \[ \log (a+\text {eps} \cosh (x)+i x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 26, normalized size = 2.17 \[ -x + \log \left (\frac {\mathit {eps} e^{\left (2 \, x\right )} + 2 \, {\left (a + i \, x\right )} e^{x} + \mathit {eps}}{\mathit {eps}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.37, size = 23, normalized size = 1.92 \[ -x + \log \left (\mathit {eps} e^{\left (2 \, x\right )} + 2 \, a e^{x} + 2 i \, x e^{x} + \mathit {eps}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 1.33 \[ \ln \left (i \mathit {eps} \cosh \relax (x )+i a -x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 13, normalized size = 1.08 \[ \log \left (i \, \mathit {eps} \cosh \relax (x) + i \, a - x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 13, normalized size = 1.08 \[ \ln \left (x-a\,1{}\mathrm {i}-\mathrm {eps}\,\mathrm {cosh}\relax (x)\,1{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.39, size = 22, normalized size = 1.83 \[ - x + \log {\left (e^{2 x} + 1 + \frac {\left (2 a + 2 i x\right ) e^{x}}{eps} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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