Optimal. Leaf size=6 \[ -\tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.02, 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 = {2249, 207} \[ -\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{-1+e^{2 x}} \, dx &=\operatorname {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 \[ -\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x}{-1+e^{2 x}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.88, size = 15, normalized size = 2.50 \[ -\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.79, size = 16, normalized size = 2.67 \[ -\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, 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] time = 0.43, size = 15, normalized size = 2.50 \[ -\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]
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 \[ \frac {\ln \left ({\mathrm {e}}^x-1\right )}{2}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 15, normalized size = 2.50 \[ \frac {\log {\left (e^{x} - 1 \right )}}{2} - \frac {\log {\left (e^{x} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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