Optimal. Leaf size=31 \[ \frac {x}{\left (2+x+e^{x+\frac {\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} x\right )^2} \]
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Rubi [F] time = 34.19, 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 {4-2 x+e^2 \left (2 x-x^2\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (-2 x-4 x^2+e^2 \left (-x^2-2 x^3\right )+8 \log \left (\frac {2+e^2 x}{x}\right )+\left (4+2 e^2 x\right ) \log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{16+24 x+12 x^2+2 x^3+e^2 \left (8 x+12 x^2+6 x^3+x^4\right )+e^{\frac {3 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (2 x^3+e^2 x^4\right )+e^{\frac {2 \left (x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )\right )}{x}} \left (12 x^2+6 x^3+e^2 \left (6 x^3+3 x^4\right )\right )+e^{\frac {x^2+\log ^2\left (\frac {2+e^2 x}{x}\right )}{x}} \left (24 x+24 x^2+6 x^3+e^2 \left (12 x^2+12 x^3+3 x^4\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\left (\left (2+e^2 x\right ) \left (-2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x+2 e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x^2\right )\right )+8 e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} \log \left (e^2+\frac {2}{x}\right )+2 e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx\\ &=\int \left (\frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2}+\frac {2 \left (4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3}\right ) \, dx\\ &=2 \int \frac {4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx\\ &=2 \int \frac {x \left (2+e^2 x\right ) \left (2+2 x+x^2\right )-4 (2+x) \log \left (e^2+\frac {2}{x}\right )-(2+x) \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {-x (1+2 x) \left (2+e^2 x\right )+8 \log \left (e^2+\frac {2}{x}\right )+2 \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx\\ &=2 \int \left (\frac {4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{2 x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3}+\frac {e^2 \left (-4 x-4 \left (1+\frac {e^2}{2}\right ) x^2-2 \left (1+e^2\right ) x^3-e^2 x^4+8 \log \left (e^2+\frac {2}{x}\right )+4 x \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )+e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )\right )}{2 \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3}\right ) \, dx+\int \left (\frac {e^2 \left (2 x+4 \left (1+\frac {e^2}{4}\right ) x^2+2 e^2 x^3-8 \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )\right )}{2 \left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2}+\frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{2 x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-2 x-4 \left (1+\frac {e^2}{4}\right ) x^2-2 e^2 x^3+8 \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+\frac {1}{2} e^2 \int \frac {2 x+4 \left (1+\frac {e^2}{4}\right ) x^2+2 e^2 x^3-8 \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 e^2 x \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+e^2 \int \frac {-4 x-4 \left (1+\frac {e^2}{2}\right ) x^2-2 \left (1+e^2\right ) x^3-e^2 x^4+8 \log \left (e^2+\frac {2}{x}\right )+4 x \log \left (e^2+\frac {2}{x}\right )+4 \log ^2\left (e^2+\frac {2}{x}\right )+2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )+e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {4 x+4 \left (1+\frac {e^2}{2}\right ) x^2+2 \left (1+e^2\right ) x^3+e^2 x^4-8 \log \left (e^2+\frac {2}{x}\right )-4 x \log \left (e^2+\frac {2}{x}\right )-4 \log ^2\left (e^2+\frac {2}{x}\right )-2 \left (1+e^2\right ) x \log ^2\left (e^2+\frac {2}{x}\right )-e^2 x^2 \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx\\ &=\frac {1}{2} \int \frac {-x (1+2 x) \left (2+e^2 x\right )+8 \log \left (e^2+\frac {2}{x}\right )+2 \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+\frac {1}{2} e^2 \int \frac {x (1+2 x) \left (2+e^2 x\right )-8 \log \left (e^2+\frac {2}{x}\right )-2 \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \, dx+e^2 \int \frac {-x \left (2+e^2 x\right ) \left (2+2 x+x^2\right )+4 (2+x) \log \left (e^2+\frac {2}{x}\right )+(2+x) \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{\left (2+e^2 x\right ) \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx+\int \frac {x \left (2+e^2 x\right ) \left (2+2 x+x^2\right )-4 (2+x) \log \left (e^2+\frac {2}{x}\right )-(2+x) \left (2+e^2 x\right ) \log ^2\left (e^2+\frac {2}{x}\right )}{x \left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 29, normalized size = 0.94 \begin {gather*} \frac {x}{\left (2+x+e^{x+\frac {\log ^2\left (e^2+\frac {2}{x}\right )}{x}} x\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 70, normalized size = 2.26 \begin {gather*} \frac {x}{x^{2} e^{\left (\frac {2 \, {\left (x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}\right )}}{x}\right )} + x^{2} + 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {x^{2} + \log \left (\frac {x e^{2} + 2}{x}\right )^{2}}{x}\right )} + 4 \, x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 32, normalized size = 1.03
method | result | size |
risch | \(\frac {x}{\left (x \,{\mathrm e}^{\frac {\ln \left (\frac {{\mathrm e}^{2} x +2}{x}\right )^{2}+x^{2}}{x}}+x +2\right )^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 5.96, size = 123, normalized size = 3.97 \begin {gather*} \frac {x e^{\left (\frac {4 \, \log \left (x e^{2} + 2\right ) \log \relax (x)}{x}\right )}}{x^{2} e^{\left (2 \, x + \frac {2 \, \log \left (x e^{2} + 2\right )^{2}}{x} + \frac {2 \, \log \relax (x)^{2}}{x}\right )} + 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x + \frac {\log \left (x e^{2} + 2\right )^{2}}{x} + \frac {2 \, \log \left (x e^{2} + 2\right ) \log \relax (x)}{x} + \frac {\log \relax (x)^{2}}{x}\right )} + {\left (x^{2} + 4 \, x + 4\right )} e^{\left (\frac {4 \, \log \left (x e^{2} + 2\right ) \log \relax (x)}{x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.34, size = 452, normalized size = 14.58 \begin {gather*} -\frac {{\left ({\mathrm {e}}^2\,x^2+2\,x\right )}^2\,\left (4\,x-8\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )-4\,x\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )-2\,x\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2+2\,x^2\,{\mathrm {e}}^2+2\,x^3\,{\mathrm {e}}^2+x^4\,{\mathrm {e}}^2-4\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2+4\,x^2+2\,x^3-x^2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2-2\,x\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2\right )}{\left (x\,{\mathrm {e}}^2+2\right )\,\left ({\left (x+2\right )}^2+x^2\,{\mathrm {e}}^{2\,x+\frac {2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2}{x}}+2\,x\,{\mathrm {e}}^{x+\frac {{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2}{x}}\,\left (x+2\right )\right )\,\left (16\,x\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )+8\,x\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2+8\,x^2\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )-8\,x^3\,{\mathrm {e}}^2-8\,x^4\,{\mathrm {e}}^2-4\,x^5\,{\mathrm {e}}^2-2\,x^4\,{\mathrm {e}}^4-2\,x^5\,{\mathrm {e}}^4-x^6\,{\mathrm {e}}^4+4\,x^2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2-8\,x^2-8\,x^3-4\,x^4+8\,x^2\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2+4\,x^3\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^2+2\,x^3\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^4+x^4\,{\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )}^2\,{\mathrm {e}}^4+8\,x^2\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )\,{\mathrm {e}}^2+4\,x^3\,\ln \left (\frac {x\,{\mathrm {e}}^2+2}{x}\right )\,{\mathrm {e}}^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.88, size = 61, normalized size = 1.97 \begin {gather*} \frac {x}{x^{2} e^{\frac {2 \left (x^{2} + \log {\left (\frac {x e^{2} + 2}{x} \right )}^{2}\right )}{x}} + x^{2} + 4 x + \left (2 x^{2} + 4 x\right ) e^{\frac {x^{2} + \log {\left (\frac {x e^{2} + 2}{x} \right )}^{2}}{x}} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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