Optimal. Leaf size=28 \[ e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right ) \]
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Rubi [A] time = 0.21, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2210, 2209, 2554, 12, 2547, 6742, 2538} \[ e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2210
Rule 2538
Rule 2547
Rule 2554
Rule 6742
Rubi steps
\begin {align*} \int \left (\frac {e^{x^2}}{x}+2 e^{x^2} x \log (x)+\frac {-2+\log (x)}{\left (x+\log ^2(x)\right )^2}+\frac {1+\frac {1}{x}+\frac {2 \log (x)}{x}}{x+\log ^2(x)}\right ) \, dx &=2 \int e^{x^2} x \log (x) \, dx+\int \frac {e^{x^2}}{x} \, dx+\int \frac {-2+\log (x)}{\left (x+\log ^2(x)\right )^2} \, dx+\int \frac {1+\frac {1}{x}+\frac {2 \log (x)}{x}}{x+\log ^2(x)} \, dx\\ &=\frac {\text {Ei}\left (x^2\right )}{2}+e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}-2 \int \frac {e^{x^2}}{2 x} \, dx-\int \frac {1}{x \left (x+\log ^2(x)\right )} \, dx+\int \left (\frac {1}{x+\log ^2(x)}+\frac {1}{x \left (x+\log ^2(x)\right )}+\frac {2 \log (x)}{x \left (x+\log ^2(x)\right )}\right ) \, dx\\ &=\frac {\text {Ei}\left (x^2\right )}{2}+e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}+2 \int \frac {\log (x)}{x \left (x+\log ^2(x)\right )} \, dx-\int \frac {e^{x^2}}{x} \, dx+\int \frac {1}{x+\log ^2(x)} \, dx\\ &=e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 28, normalized size = 1.00 \[ e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 44, normalized size = 1.57 \[ \frac {e^{\left (x^{2}\right )} \log \relax (x)^{3} + {\left (\log \relax (x)^{2} + x\right )} \log \left (\log \relax (x)^{2} + x\right ) + {\left (x e^{\left (x^{2}\right )} - 1\right )} \log \relax (x)}{\log \relax (x)^{2} + x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 27, normalized size = 0.96 \[ e^{\left (x^{2}\right )} \log \relax (x) - \frac {3 \, \log \relax (x)}{\log \relax (x)^{2} + x} + \log \left (\log \relax (x)^{2} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 1.00 \[ {\mathrm e}^{x^{2}} \ln \relax (x )-\frac {\ln \relax (x )}{\ln \relax (x )^{2}+x}+\ln \left (\ln \relax (x )^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 27, normalized size = 0.96 \[ e^{\left (x^{2}\right )} \log \relax (x) - \frac {\log \relax (x)}{\log \relax (x)^{2} + x} + \log \left (\log \relax (x)^{2} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 27, normalized size = 0.96 \[ \ln \left ({\ln \relax (x)}^2+x\right )+{\mathrm {e}}^{x^2}\,\ln \relax (x)-\frac {\ln \relax (x)}{{\ln \relax (x)}^2+x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 26, normalized size = 0.93 \[ e^{x^{2}} \log {\relax (x )} + \log {\left (x + \log {\relax (x )}^{2} \right )} - \frac {\log {\relax (x )}}{x + \log {\relax (x )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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