Optimal. Leaf size=6 \[ -\tanh ^{-1}\left (e^x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2281, 213}
\begin {gather*} -\tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 2281
Rubi steps
\begin {align*} \int \frac {e^x}{-1+e^{2 x}} \, dx &=\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^x\right )\\ &=-\tanh ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 6, normalized size = 1.00
method | result | size |
default | \(-\arctanh \left ({\mathrm e}^{x}\right )\) | \(6\) |
norman | \(\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}\) | \(16\) |
risch | \(\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (5) = 10\).
time = 1.74, size = 15, normalized size = 2.50 \begin {gather*} -\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. 15 vs.
\(2 (5) = 10\).
time = 0.80, size = 15, normalized size = 2.50 \begin {gather*} -\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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (5) = 10\).
time = 0.04, size = 15, normalized size = 2.50 \begin {gather*} \frac {\log {\left (e^{x} - 1 \right )}}{2} - \frac {\log {\left (e^{x} + 1 \right )}}{2} \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. 16 vs.
\(2 (5) = 10\).
time = 0.66, size = 16, normalized size = 2.67 \begin {gather*} -\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 = 0.00, size = 15, normalized size = 2.50 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^x-1\right )}{2}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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