Optimal. Leaf size=12 \[ \log \left (e^{-x}-e^x\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2282, 446, 72} \[ \log \left (1-e^{2 x}\right )-x \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 2282
Rubi steps
\begin {align*} \int \frac {e^{-x}+e^x}{-e^{-x}+e^x} \, dx &=\operatorname {Subst}\left (\int \frac {-1-x^2}{x \left (1-x^2\right )} \, dx,x,e^x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1-x}{(1-x) x} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {2}{-1+x}-\frac {1}{x}\right ) \, dx,x,e^{2 x}\right )\\ &=-x+\log \left (1-e^{2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.17 \[ \log \left (1-e^{2 x}\right )-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 11, normalized size = 0.92 \[ -x + \log \left (e^{\left (2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 12, normalized size = 1.00 \[ -x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 1.42 \[ \ln \left ({\mathrm e}^{x}-1\right )+\ln \left ({\mathrm e}^{x}+1\right )-\ln \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 10, normalized size = 0.83 \[ \log \left (e^{\left (-x\right )} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 11, normalized size = 0.92 \[ \ln \left ({\mathrm {e}}^{2\,x}-1\right )-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 8, normalized size = 0.67 \[ - x + \log {\left (e^{2 x} - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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