Optimal. Leaf size=19 \[ i \tanh ^{-1}(\cosh (x))+\frac {\cosh (x)}{i+\sinh (x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2826, 2727,
3855} \begin {gather*} \frac {\cosh (x)}{\sinh (x)+i}+i \tanh ^{-1}(\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2826
Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{i+\sinh (x)} \, dx &=-(i \int \text {csch}(x) \, dx)+i \int \frac {1}{i+\sinh (x)} \, dx\\ &=i \tanh ^{-1}(\cosh (x))+\frac {\cosh (x)}{i+\sinh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 1.58 \begin {gather*} \text {sech}(x) \left (-i+i \tanh ^{-1}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)}+\sinh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 21, normalized size = 1.11
method | result | size |
default | \(-i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {2}{\tanh \left (\frac {x}{2}\right )+i}\) | \(21\) |
risch | \(-\frac {2 i}{{\mathrm e}^{x}+i}+i \ln \left ({\mathrm e}^{x}+1\right )-i \ln \left ({\mathrm e}^{x}-1\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 29, normalized size = 1.53 \begin {gather*} -\frac {2 i}{e^{\left (-x\right )} - i} + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \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. 33 vs. \(2 (15) = 30\).
time = 0.46, size = 33, normalized size = 1.74 \begin {gather*} \frac {{\left (i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + {\left (-i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i}{e^{x} + i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 24, normalized size = 1.26 \begin {gather*} -\frac {2 i}{e^{x} + i} + i \, \log \left (e^{x} + 1\right ) - i \, \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 = 0.49, size = 35, normalized size = 1.84 \begin {gather*} -\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}-2{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left ({\mathrm {e}}^x\,2{}\mathrm {i}+2{}\mathrm {i}\right )\,1{}\mathrm {i}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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