Optimal. Leaf size=25 \[ \frac {e^5 \left (2+\frac {x}{x+e^x x}\right )}{2-\log (x)} \]
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Rubi [F] time = 1.64, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3 e^5+2 e^{5+2 x}+e^{5+x} (5-2 x)+e^{5+x} x \log (x)}{4 x+8 e^x x+4 e^{2 x} x+\left (-4 x-8 e^x x-4 e^{2 x} x\right ) \log (x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^5 \left (3+2 e^{2 x}+e^x (5-2 x)+e^x x \log (x)\right )}{\left (1+e^x\right )^2 x (2-\log (x))^2} \, dx\\ &=e^5 \int \frac {3+2 e^{2 x}+e^x (5-2 x)+e^x x \log (x)}{\left (1+e^x\right )^2 x (2-\log (x))^2} \, dx\\ &=e^5 \int \left (\frac {2}{x (-2+\log (x))^2}-\frac {1}{\left (1+e^x\right )^2 (-2+\log (x))}+\frac {1-2 x+x \log (x)}{\left (1+e^x\right ) x (-2+\log (x))^2}\right ) \, dx\\ &=-\left (e^5 \int \frac {1}{\left (1+e^x\right )^2 (-2+\log (x))} \, dx\right )+e^5 \int \frac {1-2 x+x \log (x)}{\left (1+e^x\right ) x (-2+\log (x))^2} \, dx+\left (2 e^5\right ) \int \frac {1}{x (-2+\log (x))^2} \, dx\\ &=-\left (e^5 \int \frac {1}{\left (1+e^x\right )^2 (-2+\log (x))} \, dx\right )+e^5 \int \left (-\frac {2}{\left (1+e^x\right ) (-2+\log (x))^2}+\frac {1}{\left (1+e^x\right ) x (-2+\log (x))^2}+\frac {\log (x)}{\left (1+e^x\right ) (-2+\log (x))^2}\right ) \, dx+\left (2 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,-2+\log (x)\right )\\ &=\frac {2 e^5}{2-\log (x)}+e^5 \int \frac {1}{\left (1+e^x\right ) x (-2+\log (x))^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 (-2+\log (x))} \, dx+e^5 \int \frac {\log (x)}{\left (1+e^x\right ) (-2+\log (x))^2} \, dx-\left (2 e^5\right ) \int \frac {1}{\left (1+e^x\right ) (-2+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.22, size = 25, normalized size = 1.00 \begin {gather*} -\frac {e^5 \left (3+2 e^x\right )}{\left (1+e^x\right ) (-2+\log (x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 36, normalized size = 1.44 \begin {gather*} -\frac {3 \, e^{10} + 2 \, e^{\left (x + 10\right )}}{{\left (e^{5} + e^{\left (x + 5\right )}\right )} \log \relax (x) - 2 \, e^{5} - 2 \, e^{\left (x + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 28, normalized size = 1.12 \begin {gather*} -\frac {3 \, e^{5} + 2 \, e^{\left (x + 5\right )}}{e^{x} \log \relax (x) - 2 \, e^{x} + \log \relax (x) - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 23, normalized size = 0.92
method | result | size |
risch | \(-\frac {\left (3+2 \,{\mathrm e}^{x}\right ) {\mathrm e}^{5}}{\left ({\mathrm e}^{x}+1\right ) \left (\ln \relax (x )-2\right )}\) | \(23\) |
norman | \(\frac {-2 \,{\mathrm e}^{5} {\mathrm e}^{x}-3 \,{\mathrm e}^{5}}{\left (\ln \relax (x )-2\right ) \left ({\mathrm e}^{x}+1\right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 26, normalized size = 1.04 \begin {gather*} -\frac {3 \, e^{5} + 2 \, e^{\left (x + 5\right )}}{{\left (\log \relax (x) - 2\right )} e^{x} + \log \relax (x) - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 100, normalized size = 4.00 \begin {gather*} \frac {\frac {{\mathrm {e}}^5}{2}+\frac {x\,{\mathrm {e}}^5}{2}}{{\mathrm {e}}^x+1}-\frac {\frac {{\mathrm {e}}^5\,\left (2\,{\mathrm {e}}^{2\,x}+5\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+3\right )}{{\left ({\mathrm {e}}^x+1\right )}^2}+\frac {x\,{\mathrm {e}}^{x+5}\,\ln \relax (x)}{{\left ({\mathrm {e}}^x+1\right )}^2}}{\ln \relax (x)-2}-\frac {\frac {{\mathrm {e}}^5\,\left (x+1\right )}{2}-{\mathrm {e}}^{x+5}\,\left (\frac {x}{2}-\frac {1}{2}\right )}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 27, normalized size = 1.08 \begin {gather*} - \frac {e^{5}}{\left (\log {\relax (x )} - 2\right ) e^{x} + \log {\relax (x )} - 2} - \frac {2 e^{5}}{\log {\relax (x )} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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