Optimal. Leaf size=27 \[ \left (x+\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )^2 \]
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Rubi [F] time = 1.26, 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 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {\int \frac {2 \left (2 \log ^2(5)+x \left (1+2 x+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \left (x+\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {2 \int \frac {\left (2 \log ^2(5)+x \left (1+2 x+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \left (x+\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {2 \int \left (\frac {2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )}+\frac {\left (2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}\\ &=\frac {2 \int \frac {2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}+\frac {2 \int \frac {\left (2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {2 \int \left (x \left (1+2 x+\log ^2(5)\right )+\frac {2 \log ^2(5)}{\log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}+\frac {2 \int \frac {\left (2 \log ^2(5)+x \left (1+2 x+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=4 \int \frac {1}{\log \left (\frac {x^2}{4}\right )} \, dx+\frac {2 \int x \left (1+2 x+\log ^2(5)\right ) \, dx}{\log ^2(5)}+\frac {2 \int \left (2 x \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\left (1+\log ^2(5)\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 \log ^2(5) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}\\ &=4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx+\frac {(4 x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+\left (2 \left (1+\frac {1}{\log ^2(5)}\right )\right ) \int \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right ) \, dx+\frac {2 \int \left (2 x^2+x \left (1+\log ^2(5)\right )\right ) \, dx}{\log ^2(5)}+\frac {4 \int x \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right ) \, dx}{\log ^2(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )+\frac {4 x^3}{3 \log ^2(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx-\left (2 \left (1+\frac {1}{\log ^2(5)}\right )\right ) \int \left (\frac {x (1+2 x)}{\log ^2(5)}+\frac {2}{\log \left (\frac {x^2}{4}\right )}\right ) \, dx-\frac {4 \int \left (\frac {x^2 (1+2 x)}{2 \log ^2(5)}+\frac {x}{\log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )+\frac {4 x^3}{3 \log ^2(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx-\left (4 \left (1+\frac {1}{\log ^2(5)}\right )\right ) \int \frac {1}{\log \left (\frac {x^2}{4}\right )} \, dx-\frac {2 \int x^2 (1+2 x) \, dx}{\log ^4(5)}-\frac {4 \int \frac {x}{\log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}-\frac {\left (2 \left (1+\log ^2(5)\right )\right ) \int x (1+2 x) \, dx}{\log ^4(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )+\frac {4 x^3}{3 \log ^2(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx-\frac {\left (4 x \left (1+\frac {1}{\log ^2(5)}\right )\right ) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}-\frac {2 \int \left (x^2+2 x^3\right ) \, dx}{\log ^4(5)}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {x}{4}\right )} \, dx,x,x^2\right )}{\log ^2(5)}-\frac {\left (2 \left (1+\log ^2(5)\right )\right ) \int \left (x+2 x^2\right ) \, dx}{\log ^4(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )-\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right ) \left (1+\frac {1}{\log ^2(5)}\right )}{\sqrt {x^2}}-\frac {2 x^3}{3 \log ^4(5)}-\frac {x^4}{\log ^4(5)}+\frac {4 x^3}{3 \log ^2(5)}-\frac {x^2 \left (1+\log ^2(5)\right )}{\log ^4(5)}-\frac {4 x^3 \left (1+\log ^2(5)\right )}{3 \log ^4(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}-\frac {8 \text {li}\left (\frac {x^2}{4}\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.23, size = 85, normalized size = 3.15 \begin {gather*} -\frac {\left (x \left (1+x+\log ^2(5)\right )+\log ^2(5) \log \left (\log \left (\frac {x^2}{4}\right )\right )\right ) \left (x \left (1+x-\log ^2(5)\right )+\log ^2(5) \log \left (\log \left (\frac {x^2}{4}\right )\right )-2 \log ^2(5) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{\log ^4(5)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 49, normalized size = 1.81 \begin {gather*} x^{2} + 2 \, x \log \left (e^{\left (\frac {x^{2} + x - 1}{\log \relax (5)^{2}}\right )} \log \left (\frac {1}{4} \, x^{2}\right )\right ) + \log \left (e^{\left (\frac {x^{2} + x - 1}{\log \relax (5)^{2}}\right )} \log \left (\frac {1}{4} \, x^{2}\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.72, size = 132, normalized size = 4.89 \begin {gather*} -\frac {\log \relax (5)^{2} \log \left (-2 \, \log \relax (2) + \log \left (x^{2}\right )\right )^{2} - \frac {2 \, {\left (\log \relax (5)^{2} + 1\right )} x^{3}}{\log \relax (5)^{2}} - \frac {x^{4}}{\log \relax (5)^{2}} - 2 \, {\left (\log \relax (5)^{2} \log \left (-2 \, \log \relax (2) + \log \left (x^{2}\right )\right ) + {\left (\log \relax (5)^{2} + 1\right )} x + x^{2}\right )} \log \left (\log \left (\frac {1}{4} \, x^{2}\right )\right ) - \frac {{\left (\log \relax (5)^{4} + 2 \, \log \relax (5)^{2} - 1\right )} x^{2}}{\log \relax (5)^{2}} + \frac {2 \, {\left (\log \relax (5)^{2} + 1\right )} x}{\log \relax (5)^{2}} + 2 \, \log \left (2 \, \log \relax (2) - \log \left (x^{2}\right )\right )}{\log \relax (5)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.45, size = 4657, normalized size = 172.48
method | result | size |
risch | \(\text {Expression too large to display}\) | \(4657\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 148, normalized size = 5.48 \begin {gather*} \frac {3 \, x^{2} \log \relax (5)^{2} + 4 \, x^{3} + 3 \, x^{2} + \frac {3 \, \log \relax (5)^{4} \log \left (-\log \relax (2) + \log \relax (x)\right )^{2} + 2 \, {\left (\log \relax (5)^{2} + 3\right )} x^{3} + 3 \, x^{4} + 3 \, {\left (2 \, \log \relax (5)^{2} \log \relax (2) + \log \relax (5)^{2} - 1\right )} x^{2} - 6 \, {\left (\log \relax (5)^{2} - {\left (\log \relax (5)^{4} + \log \relax (5)^{2}\right )} \log \relax (2) + 1\right )} x + 6 \, {\left (\log \relax (5)^{4} \log \relax (2) + x^{2} \log \relax (5)^{2} + {\left (\log \relax (5)^{4} + \log \relax (5)^{2}\right )} x - \log \relax (5)^{2}\right )} \log \left (-\log \relax (2) + \log \relax (x)\right )}{\log \relax (5)^{2}}}{3 \, \log \relax (5)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.83, size = 35, normalized size = 1.30 \begin {gather*} {\left (x+\ln \left ({\mathrm {e}}^{\frac {x}{{\ln \relax (5)}^2}}\,{\mathrm {e}}^{\frac {x^2}{{\ln \relax (5)}^2}}\,{\mathrm {e}}^{-\frac {1}{{\ln \relax (5)}^2}}\,\ln \left (\frac {x^2}{4}\right )\right )\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.67, size = 51, normalized size = 1.89 \begin {gather*} x^{2} + 2 x \log {\left (e^{\frac {x^{2} + x - 1}{\log {\relax (5 )}^{2}}} \log {\left (\frac {x^{2}}{4} \right )} \right )} + \log {\left (e^{\frac {x^{2} + x - 1}{\log {\relax (5 )}^{2}}} \log {\left (\frac {x^{2}}{4} \right )} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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