Optimal. Leaf size=12 \[ \log (a+i x+\text {eps} \cosh (x)) \]
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Rubi [A]
time = 0.02, 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 = {6816}
\begin {gather*} \log (a+\text {eps} \cosh (x)+i x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
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 = 0.04, size = 12, normalized size = 1.00 \begin {gather*} \log (a+i x+\text {eps} \cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 16, normalized size = 1.33
method | result | size |
derivativedivides | \(\ln \left (i a -x +i \mathit {eps} \cosh \left (x \right )\right )\) | \(16\) |
default | \(\ln \left (i a -x +i \mathit {eps} \cosh \left (x \right )\right )\) | \(16\) |
risch | \(-x +\ln \left (1+\frac {2 \left (i x +a \right ) {\mathrm e}^{x}}{\mathit {eps}}+{\mathrm e}^{2 x}\right )\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.70, size = 13, normalized size = 1.08 \begin {gather*} \log \left (i \, \mathit {eps} \cosh \left (x\right ) + i \, a - x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 26 vs. \(2 (10) = 20\).
time = 0.83, size = 26, normalized size = 2.17 \begin {gather*} -x + \log \left (\frac {\mathit {eps} e^{\left (2 \, x\right )} + 2 \, {\left (a + i \, x\right )} e^{x} + \mathit {eps}}{\mathit {eps}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 22 vs. \(2 (10) = 20\).
time = 0.15, size = 22, normalized size = 1.83 \begin {gather*} - x + \log {\left (e^{2 x} + 1 + \frac {\left (2 a + 2 i x\right ) e^{x}}{eps} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 23 vs. \(2 (10) = 20\).
time = 0.49, size = 23, normalized size = 1.92 \begin {gather*} -x + \log \left (\mathit {eps} e^{\left (2 \, x\right )} + 2 \, a e^{x} + 2 i \, x e^{x} + \mathit {eps}\right ) \end {gather*}
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 \begin {gather*} \ln \left (x-a\,1{}\mathrm {i}-\mathrm {eps}\,\mathrm {cosh}\left (x\right )\,1{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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