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 = 0.15, 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 = {2241, 2240,
2634, 12, 2627, 6874, 2618} \begin {gather*} e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2240
Rule 2241
Rule 2618
Rule 2627
Rule 2634
Rule 6874
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 = 40.00, size = 28, normalized size = 1.00 \begin {gather*} e^{x^2} \log (x)-\frac {\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 28, normalized size = 1.00
method | result | size |
default | \({\mathrm e}^{x^{2}} \ln \left (x \right )-\frac {\ln \left (x \right )}{x +\ln \left (x \right )^{2}}+\ln \left (x +\ln \left (x \right )^{2}\right )\) | \(28\) |
risch | \({\mathrm e}^{x^{2}} \ln \left (x \right )-\frac {\ln \left (x \right )}{x +\ln \left (x \right )^{2}}+\ln \left (x +\ln \left (x \right )^{2}\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.93, size = 27, normalized size = 0.96 \begin {gather*} e^{\left (x^{2}\right )} \log \left (x\right ) - \frac {\log \left (x\right )}{\log \left (x\right )^{2} + x} + \log \left (\log \left (x\right )^{2} + x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.85, size = 44, normalized size = 1.57 \begin {gather*} \frac {e^{\left (x^{2}\right )} \log \left (x\right )^{3} + {\left (\log \left (x\right )^{2} + x\right )} \log \left (\log \left (x\right )^{2} + x\right ) + {\left (x e^{\left (x^{2}\right )} - 1\right )} \log \left (x\right )}{\log \left (x\right )^{2} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 26, normalized size = 0.93 \begin {gather*} e^{x^{2}} \log {\left (x \right )} + \log {\left (x + \log {\left (x \right )}^{2} \right )} - \frac {\log {\left (x \right )}}{x + \log {\left (x \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 27, normalized size = 0.96 \begin {gather*} e^{\left (x^{2}\right )} \log \left (x\right ) - \frac {3 \, \log \left (x\right )}{\log \left (x\right )^{2} + x} + \log \left (\log \left (x\right )^{2} + x\right ) \end {gather*}
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 \begin {gather*} \ln \left ({\ln \left (x\right )}^2+x\right )+{\mathrm {e}}^{x^2}\,\ln \left (x\right )-\frac {\ln \left (x\right )}{{\ln \left (x\right )}^2+x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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