Optimal. Leaf size=16 \[ \frac {e^2 (-1+x)^2}{2 \log ^2(x)} \]
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Rubi [B] time = 0.36, antiderivative size = 71, normalized size of antiderivative = 4.44, number of steps used = 22, number of rules used = 13, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {6688, 12, 6742, 2353, 2297, 2298, 2302, 30, 2306, 2309, 2178, 2320, 2330} \begin {gather*} \frac {e^2 x^2}{2 \log ^2(x)}+\frac {e^2 x^2}{\log (x)}-\frac {e^2 x}{\log ^2(x)}+\frac {e^2}{2 \log ^2(x)}+\frac {e^2 (1-x) x}{\log (x)}-\frac {e^2 x}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2178
Rule 2297
Rule 2298
Rule 2302
Rule 2306
Rule 2309
Rule 2320
Rule 2330
Rule 2353
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^2 (1-x) (-1+x-x \log (x))}{x \log ^3(x)} \, dx\\ &=e^2 \int \frac {(1-x) (-1+x-x \log (x))}{x \log ^3(x)} \, dx\\ &=e^2 \int \left (-\frac {(-1+x)^2}{x \log ^3(x)}+\frac {-1+x}{\log ^2(x)}\right ) \, dx\\ &=-\left (e^2 \int \frac {(-1+x)^2}{x \log ^3(x)} \, dx\right )+e^2 \int \frac {-1+x}{\log ^2(x)} \, dx\\ &=\frac {e^2 (1-x) x}{\log (x)}-e^2 \int \left (-\frac {2}{\log ^3(x)}+\frac {1}{x \log ^3(x)}+\frac {x}{\log ^3(x)}\right ) \, dx+e^2 \int \frac {1}{\log (x)} \, dx+\left (2 e^2\right ) \int \frac {-1+x}{\log (x)} \, dx\\ &=\frac {e^2 (1-x) x}{\log (x)}+e^2 \text {li}(x)-e^2 \int \frac {1}{x \log ^3(x)} \, dx-e^2 \int \frac {x}{\log ^3(x)} \, dx+\left (2 e^2\right ) \int \left (-\frac {1}{\log (x)}+\frac {x}{\log (x)}\right ) \, dx+\left (2 e^2\right ) \int \frac {1}{\log ^3(x)} \, dx\\ &=-\frac {e^2 x}{\log ^2(x)}+\frac {e^2 x^2}{2 \log ^2(x)}+\frac {e^2 (1-x) x}{\log (x)}+e^2 \text {li}(x)+e^2 \int \frac {1}{\log ^2(x)} \, dx-e^2 \int \frac {x}{\log ^2(x)} \, dx-e^2 \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (x)\right )-\left (2 e^2\right ) \int \frac {1}{\log (x)} \, dx+\left (2 e^2\right ) \int \frac {x}{\log (x)} \, dx\\ &=\frac {e^2}{2 \log ^2(x)}-\frac {e^2 x}{\log ^2(x)}+\frac {e^2 x^2}{2 \log ^2(x)}-\frac {e^2 x}{\log (x)}+\frac {e^2 (1-x) x}{\log (x)}+\frac {e^2 x^2}{\log (x)}-e^2 \text {li}(x)+e^2 \int \frac {1}{\log (x)} \, dx-\left (2 e^2\right ) \int \frac {x}{\log (x)} \, dx+\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=2 e^2 \text {Ei}(2 \log (x))+\frac {e^2}{2 \log ^2(x)}-\frac {e^2 x}{\log ^2(x)}+\frac {e^2 x^2}{2 \log ^2(x)}-\frac {e^2 x}{\log (x)}+\frac {e^2 (1-x) x}{\log (x)}+\frac {e^2 x^2}{\log (x)}-\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {e^2}{2 \log ^2(x)}-\frac {e^2 x}{\log ^2(x)}+\frac {e^2 x^2}{2 \log ^2(x)}-\frac {e^2 x}{\log (x)}+\frac {e^2 (1-x) x}{\log (x)}+\frac {e^2 x^2}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 16, normalized size = 1.00 \begin {gather*} \frac {e^2 (-1+x)^2}{2 \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 16, normalized size = 1.00 \begin {gather*} \frac {{\left (x^{2} - 2 \, x + 1\right )} e^{2}}{2 \, \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 20, normalized size = 1.25 \begin {gather*} \frac {x^{2} e^{2} - 2 \, x e^{2} + e^{2}}{2 \, \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 17, normalized size = 1.06
method | result | size |
risch | \(\frac {{\mathrm e}^{2} \left (x^{2}-2 x +1\right )}{2 \ln \relax (x )^{2}}\) | \(17\) |
norman | \(\frac {\frac {{\mathrm e}^{2}}{2}-{\mathrm e}^{2} x +\frac {x^{2} {\mathrm e}^{2}}{2}}{\ln \relax (x )^{2}}\) | \(29\) |
default | \({\mathrm e}^{2} \left (-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )-{\mathrm e}^{2} \left (-\frac {x}{\ln \relax (x )}-\expIntegralEi \left (1, -\ln \relax (x )\right )\right )-{\mathrm e}^{2} \left (-\frac {x^{2}}{2 \ln \relax (x )^{2}}-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )+2 \,{\mathrm e}^{2} \left (-\frac {x}{2 \ln \relax (x )^{2}}-\frac {x}{2 \ln \relax (x )}-\frac {\expIntegralEi \left (1, -\ln \relax (x )\right )}{2}\right )+\frac {{\mathrm e}^{2}}{2 \ln \relax (x )^{2}}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 49, normalized size = 3.06 \begin {gather*} -e^{2} \Gamma \left (-1, -\log \relax (x)\right ) + 2 \, e^{2} \Gamma \left (-1, -2 \, \log \relax (x)\right ) - 2 \, e^{2} \Gamma \left (-2, -\log \relax (x)\right ) + 4 \, e^{2} \Gamma \left (-2, -2 \, \log \relax (x)\right ) + \frac {e^{2}}{2 \, \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 13, normalized size = 0.81 \begin {gather*} \frac {{\mathrm {e}}^2\,{\left (x-1\right )}^2}{2\,{\ln \relax (x)}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 22, normalized size = 1.38 \begin {gather*} \frac {x^{2} e^{2} - 2 x e^{2} + e^{2}}{2 \log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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