Optimal. Leaf size=12 \[ e^{-x}-\tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2282, 325, 207} \[ e^{-x}-\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 325
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{-e^x+e^{3 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x^2 \left (-1+x^2\right )} \, dx,x,e^x\right )\\ &=e^{-x}+\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^x\right )\\ &=e^{-x}-\tanh ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 19, normalized size = 1.58 \[ e^{-x} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};e^{2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 25, normalized size = 2.08 \[ -\frac {1}{2} \, {\left (e^{x} \log \left (e^{x} + 1\right ) - e^{x} \log \left (e^{x} - 1\right ) - 2\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 20, normalized size = 1.67 \[ e^{\left (-x\right )} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 1.67 \[ {\mathrm e}^{-x}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 19, normalized size = 1.58 \[ e^{\left (-x\right )} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 19, normalized size = 1.58 \[ {\mathrm {e}}^{-x}+\frac {\ln \left ({\mathrm {e}}^x-1\right )}{2}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 20, normalized size = 1.67 \[ \frac {\log {\left (e^{x} - 1 \right )}}{2} - \frac {\log {\left (e^{x} + 1 \right )}}{2} + e^{- x} \]
Verification of antiderivative is not currently implemented for this CAS.
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