Optimal. Leaf size=12 \[ -\tanh ^{-1}(\cosh (x))+i \coth (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 3852, 8,
3855} \begin {gather*} -\tanh ^{-1}(\cosh (x))+i \coth (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2785
Rule 3852
Rule 3855
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{i+\sinh (x)} \, dx &=-\left (i \int \text {csch}^2(x) \, dx\right )+\int \text {csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))-\text {Subst}(\int 1 \, dx,x,-i \coth (x))\\ &=-\tanh ^{-1}(\cosh (x))+i \coth (x)\\ \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. \(32\) vs. \(2(12)=24\).
time = 0.03, size = 32, normalized size = 2.67 \begin {gather*} \frac {1}{2} i \coth \left (\frac {x}{2}\right )+\log \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{2} i \tanh \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 23, normalized size = 1.92
method | result | size |
default | \(\frac {i \tanh \left (\frac {x}{2}\right )}{2}+\frac {i}{2 \tanh \left (\frac {x}{2}\right )}+\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(23\) |
risch | \(\frac {2 i}{{\mathrm e}^{2 x}-1}+\ln \left ({\mathrm e}^{x}-1\right )-\ln \left ({\mathrm e}^{x}+1\right )\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).
time = 0.28, size = 27, normalized size = 2.25 \begin {gather*} -\frac {2 i}{e^{\left (-2 \, x\right )} - 1} - \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\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. 37 vs. \(2 (10) = 20\).
time = 0.42, size = 37, normalized size = 3.08 \begin {gather*} -\frac {{\left (e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} + 1\right ) - {\left (e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} - 1\right ) - 2 i}{e^{\left (2 \, x\right )} - 1} \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. \(2 (8) = 16\).
time = 0.06, size = 22, normalized size = 1.83 \begin {gather*} \log {\left (e^{x} - 1 \right )} - \log {\left (e^{x} + 1 \right )} + \frac {2 i}{e^{2 x} - 1} \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. 24 vs. \(2 (10) = 20\).
time = 0.42, size = 24, normalized size = 2.00 \begin {gather*} \frac {2 i}{e^{\left (2 \, x\right )} - 1} - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 28, normalized size = 2.33 \begin {gather*} \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )+\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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