Optimal. Leaf size=32 \[ \frac {x}{2}-\frac {x}{2 \left (1+e^{2 x}\right )}-\frac {1}{4} \log \left (1+e^{2 x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2321, 2222,
2320, 36, 29, 31} \begin {gather*} -\frac {x}{2 \left (e^{2 x}+1\right )}+\frac {x}{2}-\frac {1}{4} \log \left (e^{2 x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2222
Rule 2320
Rule 2321
Rubi steps
\begin {align*} \int \frac {x}{\left (e^{-x}+e^x\right )^2} \, dx &=\int \frac {e^{2 x} x}{\left (1+e^{2 x}\right )^2} \, dx\\ &=-\frac {x}{2 \left (1+e^{2 x}\right )}+\frac {1}{2} \int \frac {1}{1+e^{2 x}} \, dx\\ &=-\frac {x}{2 \left (1+e^{2 x}\right )}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^{2 x}\right )\\ &=-\frac {x}{2 \left (1+e^{2 x}\right )}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^{2 x}\right )\\ &=\frac {x}{2}-\frac {x}{2 \left (1+e^{2 x}\right )}-\frac {1}{4} \log \left (1+e^{2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 31, normalized size = 0.97 \begin {gather*} \frac {e^{2 x} x}{2+2 e^{2 x}}-\frac {1}{4} \log \left (1+e^{2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.81, size = 40, normalized size = 1.25 \begin {gather*} \frac {-2 x-\text {Log}\left [1+E^{-2 x}\right ]-\text {Log}\left [1+E^{-2 x}\right ] E^{2 x}}{4+4 E^{2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 26, normalized size = 0.81
method | result | size |
risch | \(\frac {x}{2}-\frac {x}{2 \left (1+{\mathrm e}^{2 x}\right )}-\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{4}\) | \(25\) |
default | \(-\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{4}+\frac {{\mathrm e}^{2 x} x}{2+2 \,{\mathrm e}^{2 x}}\) | \(26\) |
norman | \(-\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{4}+\frac {{\mathrm e}^{2 x} x}{2+2 \,{\mathrm e}^{2 x}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 25, normalized size = 0.78 \begin {gather*} \frac {x e^{\left (2 \, x\right )}}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} - \frac {1}{4} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 33, normalized size = 1.03 \begin {gather*} \frac {2 \, x e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{4 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 24, normalized size = 0.75 \begin {gather*} - \frac {x}{2} + \frac {x}{2 + 2 e^{- 2 x}} - \frac {\log {\left (1 + e^{- 2 x} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 39, normalized size = 1.22 \begin {gather*} \frac {2 x \mathrm {e}^{2 x}-\mathrm {e}^{2 x} \ln \left (\mathrm {e}^{2 x}+1\right )-\ln \left (\mathrm {e}^{2 x}+1\right )}{4 \mathrm {e}^{2 x}+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 26, normalized size = 0.81 \begin {gather*} \frac {x\,{\mathrm {e}}^{2\,x}}{2\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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