Optimal. Leaf size=33 \[ e^{e^4}-x^2+\left (1+x^2\right )^2 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \]
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Rubi [F] time = 0.38, 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 {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2 x-\frac {2 \left (1+x^2\right )^2 \left (1+2 x^2\right ) \log \left (\frac {e^{-x^2}}{x}\right )}{x}+4 x \left (1+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )\right ) \, dx\\ &=-x^2-2 \int \frac {\left (1+x^2\right )^2 \left (1+2 x^2\right ) \log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+4 \int x \left (1+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-x^2-2 \int \left (\frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x}+4 x \log \left (\frac {e^{-x^2}}{x}\right )+5 x^3 \log \left (\frac {e^{-x^2}}{x}\right )+2 x^5 \log \left (\frac {e^{-x^2}}{x}\right )\right ) \, dx+4 \int \left (x \log ^2\left (\frac {e^{-x^2}}{x}\right )+x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right )\right ) \, dx\\ &=-x^2-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx-4 \int x^5 \log \left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx-8 \int x \log \left (\frac {e^{-x^2}}{x}\right ) \, dx-10 \int x^3 \log \left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-x^2-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )+\frac {2}{3} \int x^5 \left (-1-2 x^2\right ) \, dx-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+\frac {5}{2} \int x^3 \left (-1-2 x^2\right ) \, dx+4 \int x \left (-1-2 x^2\right ) \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-x^2-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )+\frac {2}{3} \int \left (-x^5-2 x^7\right ) \, dx-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+\frac {5}{2} \int \left (-x^3-2 x^5\right ) \, dx+4 \int \left (-x-2 x^3\right ) \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ &=-3 x^2-\frac {21 x^4}{8}-\frac {17 x^6}{18}-\frac {x^8}{6}-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx+4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.08, size = 72, normalized size = 2.18 \begin {gather*} \log ^2\left (\frac {1}{x}\right )+x^2 \left (2+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )+2 \log \left (\frac {1}{x}\right ) \log (x)-2 \log \left (\frac {e^{-x^2}}{x}\right ) \left (x^2+\log (x)\right )-x^2 \left (1+x^2+2 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 30, normalized size = 0.91 \begin {gather*} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 48, normalized size = 1.45 \begin {gather*} x^{8} + 2 \, x^{6} + x^{4} + {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \relax (x)^{2} - x^{2} + 2 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 468, normalized size = 14.18
method | result | size |
default | \(-2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) \ln \relax (x )+\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right )^{2} x^{4}+2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right )^{2} x^{2}+\frac {2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right ) x^{6}}{3}+\frac {5 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right ) x^{4}}{2}+2 x^{2} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right )-\frac {2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{6}}{3}-\frac {5 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{4}}{2}-4 x^{2} \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )-4 x^{2} \ln \relax (x )+2 x^{2} \ln \relax (x )^{2}-\ln \relax (x )^{2}-\frac {x^{6}}{2}-3 x^{4}-x^{2}+\frac {4 x^{6} \ln \relax (x )}{3}+2 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) x^{4} \ln \relax (x )+4 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) x^{2} \ln \relax (x )+x^{4} \ln \relax (x )^{2}+\frac {3 x^{4} \ln \relax (x )}{2}-\frac {2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{6}}{3}-2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}+\ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{4}+2 \ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{2}-\frac {\left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) x^{4}}{2}-2 x^{2} \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right )-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right ) x^{4} \ln \relax (x )-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right ) x^{2} \ln \relax (x )-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right ) \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \relax (x )\right ) \ln \left ({\mathrm e}^{x^{2}}\right ) x^{2}\) | \(468\) |
risch | \(\pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3}-2 \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{4}+i \pi \,x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3} \ln \relax (x )-\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2}}{4}+\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3}}{2}+\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3}}{2}-\pi ^{2} x^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{4}-i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2}-i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) x^{2} \pi -\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2}}{2}+\pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3}-i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2}-i \pi \ln \relax (x ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2}+2 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3} \ln \relax (x )+i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right ) \ln \relax (x )+2 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right ) \ln \relax (x )+2 x^{2} \ln \relax (x )^{2}+\ln \relax (x )^{2}-x^{4}-x^{2}+x^{4} \ln \relax (x )^{2}-2 i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \ln \relax (x )-i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \ln \relax (x )-2 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \ln \relax (x )-i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \ln \relax (x )+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )+i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) x^{2} \pi +i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3} x^{2} \pi -\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{4}}{2}+\pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{5}-\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{4}}{2}+\pi ^{2} x^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{5}+i \pi \ln \relax (x ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3}-\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{4}}{4}+\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{5}}{2}-\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{4}}{4}+\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{5}}{2}+\left (x^{4}+2 x^{2}\right ) \ln \left ({\mathrm e}^{x^{2}}\right )^{2}-\frac {\pi ^{2} x^{4} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{6}}{4}-\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{6}}{2}+\left (i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) x^{4} \pi -i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) x^{4} \pi -i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) x^{4} \pi +i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3} x^{4} \pi +2 i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) x^{2} \pi -2 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2}-2 i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) x^{2} \pi +2 i \mathrm {csgn}\left (\frac {i {\mathrm e}^{-x^{2}}}{x}\right )^{3} x^{2} \pi +2 x^{4} \ln \relax (x )+4 x^{2} \ln \relax (x )+2 x^{2}+2 \ln \relax (x )\right ) \ln \left ({\mathrm e}^{x^{2}}\right )\) | \(1450\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 171, normalized size = 5.18 \begin {gather*} -\frac {2}{3} \, x^{6} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - \frac {5}{2} \, x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{4} + 2 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - 4 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{2} - {\left (2 \, x^{2} + \log \left (x^{2}\right )\right )} \log \relax (x) + \log \relax (x)^{2} + \frac {1}{6} \, {\left (4 \, x^{6} + 3 \, x^{4}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + 2 \, {\left (x^{4} + x^{2}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - 2 \, \log \relax (x) \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 30, normalized size = 0.91 \begin {gather*} {\ln \left (\frac {{\mathrm {e}}^{-x^2}}{x}\right )}^2\,\left (x^4+2\,x^2+1\right )-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 24, normalized size = 0.73 \begin {gather*} - x^{2} + \left (x^{4} + 2 x^{2} + 1\right ) \log {\left (\frac {e^{- x^{2}}}{x} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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