Optimal. Leaf size=11 \[ \text {ArcTan}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {2320, 12, 304,
209, 212} \begin {gather*} \text {ArcTan}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 212
Rule 304
Rule 2320
Rubi steps
\begin {align*} \int e^x \text {csch}(2 x) \, dx &=\text {Subst}\left (\int \frac {2 x^2}{-1+x^4} \, dx,x,e^x\right )\\ &=2 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,e^x\right )\\ &=-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (e^x\right )-\tanh ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \text {ArcTan}\left (e^x\right )-\tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.58, size = 34, normalized size = 3.09
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 18, normalized size = 1.64 \begin {gather*} \arctan \left (e^{x}\right ) - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (9) = 18\).
time = 0.39, size = 25, normalized size = 2.27 \begin {gather*} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac {1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \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 e^{x} \operatorname {csch}{\left (2 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs.
\(2 (9) = 18\).
time = 0.40, size = 19, normalized size = 1.73 \begin {gather*} \arctan \left (e^{x}\right ) - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \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 = 0.17, size = 26, normalized size = 2.36 \begin {gather*} \frac {\ln \left (4\,{\mathrm {e}}^x-4\right )}{2}-\frac {\ln \left (-4\,{\mathrm {e}}^x-4\right )}{2}-\mathrm {atan}\left ({\mathrm {e}}^{-x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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