Optimal. Leaf size=10 \[ e^x-\tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2248, 321, 207} \[ e^x-\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{3 x}}{-1+e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, 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 [A] time = 0.01, size = 10, normalized size = 1.00 \[ e^x-\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 17, normalized size = 1.70 \[ e^{x} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 18, normalized size = 1.80 \[ e^{x} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 18, normalized size = 1.80 \[ {\mathrm e}^{x}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 17, normalized size = 1.70 \[ e^{x} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 17, normalized size = 1.70 \[ \frac {\ln \left ({\mathrm {e}}^x-1\right )}{2}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{2}+{\mathrm {e}}^x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 19, normalized size = 1.90 \[ e^{x} + \frac {\log {\left (e^{x} - 1 \right )}}{2} - \frac {\log {\left (e^{x} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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