Optimal. Leaf size=12 \[ \log (a+\text{eps} \cosh (x)+i x) \]
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Rubi [A] time = 0.0325908, antiderivative size = 12, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0827899, size = 12, normalized size = 1. \[ \log (a+\text{eps} \cosh (x)+i x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 16, normalized size = 1.3 \begin{align*} \ln \left ( ia-x+i{\it eps}\,\cosh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966939, size = 18, normalized size = 1.5 \begin{align*} \log \left (i \, \mathit{eps} \cosh \left (x\right ) + i \, a - x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72218, size = 74, normalized size = 6.17 \begin{align*} -x + \log \left (\frac{\mathit{eps} e^{\left (2 \, x\right )} +{\left (2 \, a + 2 i \, x\right )} e^{x} + \mathit{eps}}{\mathit{eps}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.33109, size = 22, normalized size = 1.83 \begin{align*} - x + \log{\left (e^{2 x} + 1 + \frac{\left (2 a + 2 i x\right ) e^{x}}{eps} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08743, size = 31, normalized size = 2.58 \begin{align*} -x + \log \left (\mathit{eps} e^{\left (2 \, x\right )} + 2 \, a e^{x} + 2 i \, x e^{x} + \mathit{eps}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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