Optimal. Leaf size=20 \[ \log (2)-\left (1-\frac {1+2 x}{\log (x)}\right )^2 \]
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Rubi [A] time = 0.41, antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 23, number of rules used = 9, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {6742, 2353, 2297, 2298, 2302, 30, 2306, 2309, 2178} \begin {gather*} -\frac {4 x^2}{\log ^2(x)}-\frac {4 x}{\log ^2(x)}-\frac {1}{\log ^2(x)}+\frac {4 x}{\log (x)}+\frac {2}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2178
Rule 2297
Rule 2298
Rule 2302
Rule 2306
Rule 2309
Rule 2353
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 (1+2 x)^2}{x \log ^3(x)}-\frac {2 (1+2 x)^2}{x \log ^2(x)}+\frac {4}{\log (x)}\right ) \, dx\\ &=2 \int \frac {(1+2 x)^2}{x \log ^3(x)} \, dx-2 \int \frac {(1+2 x)^2}{x \log ^2(x)} \, dx+4 \int \frac {1}{\log (x)} \, dx\\ &=4 \text {li}(x)+2 \int \left (\frac {4}{\log ^3(x)}+\frac {1}{x \log ^3(x)}+\frac {4 x}{\log ^3(x)}\right ) \, dx-2 \int \left (\frac {4}{\log ^2(x)}+\frac {1}{x \log ^2(x)}+\frac {4 x}{\log ^2(x)}\right ) \, dx\\ &=4 \text {li}(x)+2 \int \frac {1}{x \log ^3(x)} \, dx-2 \int \frac {1}{x \log ^2(x)} \, dx+8 \int \frac {1}{\log ^3(x)} \, dx+8 \int \frac {x}{\log ^3(x)} \, dx-8 \int \frac {1}{\log ^2(x)} \, dx-8 \int \frac {x}{\log ^2(x)} \, dx\\ &=-\frac {4 x}{\log ^2(x)}-\frac {4 x^2}{\log ^2(x)}+\frac {8 x}{\log (x)}+\frac {8 x^2}{\log (x)}+4 \text {li}(x)+2 \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (x)\right )-2 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )+4 \int \frac {1}{\log ^2(x)} \, dx+8 \int \frac {x}{\log ^2(x)} \, dx-8 \int \frac {1}{\log (x)} \, dx-16 \int \frac {x}{\log (x)} \, dx\\ &=-\frac {1}{\log ^2(x)}-\frac {4 x}{\log ^2(x)}-\frac {4 x^2}{\log ^2(x)}+\frac {2}{\log (x)}+\frac {4 x}{\log (x)}-4 \text {li}(x)+4 \int \frac {1}{\log (x)} \, dx+16 \int \frac {x}{\log (x)} \, dx-16 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-16 \text {Ei}(2 \log (x))-\frac {1}{\log ^2(x)}-\frac {4 x}{\log ^2(x)}-\frac {4 x^2}{\log ^2(x)}+\frac {2}{\log (x)}+\frac {4 x}{\log (x)}+16 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {1}{\log ^2(x)}-\frac {4 x}{\log ^2(x)}-\frac {4 x^2}{\log ^2(x)}+\frac {2}{\log (x)}+\frac {4 x}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 20, normalized size = 1.00 \begin {gather*} -\frac {(1+2 x) (1+2 x-2 \log (x))}{\log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 25, normalized size = 1.25 \begin {gather*} -\frac {4 \, x^{2} - 2 \, {\left (2 \, x + 1\right )} \log \relax (x) + 4 \, x + 1}{\log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 25, normalized size = 1.25 \begin {gather*} -\frac {4 \, x^{2} - 4 \, x \log \relax (x) + 4 \, x - 2 \, \log \relax (x) + 1}{\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 = 25, normalized size = 1.25
method | result | size |
norman | \(\frac {-1-4 x -4 x^{2}+4 x \ln \relax (x )+2 \ln \relax (x )}{\ln \relax (x )^{2}}\) | \(25\) |
risch | \(-\frac {4 x^{2}-4 x \ln \relax (x )+4 x -2 \ln \relax (x )+1}{\ln \relax (x )^{2}}\) | \(26\) |
default | \(\frac {4 x}{\ln \relax (x )}-\frac {4 x^{2}}{\ln \relax (x )^{2}}+\frac {2}{\ln \relax (x )}-\frac {4 x}{\ln \relax (x )^{2}}-\frac {1}{\ln \relax (x )^{2}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 50, normalized size = 2.50 \begin {gather*} \frac {2}{\log \relax (x)} - \frac {1}{\log \relax (x)^{2}} + 4 \, {\rm Ei}\left (\log \relax (x)\right ) - 8 \, \Gamma \left (-1, -\log \relax (x)\right ) - 16 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) - 8 \, \Gamma \left (-2, -\log \relax (x)\right ) - 32 \, \Gamma \left (-2, -2 \, \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.12, size = 24, normalized size = 1.20 \begin {gather*} \frac {4\,x+2}{\ln \relax (x)}-\frac {{\left (2\,x+1\right )}^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 = 1.10 \begin {gather*} \frac {- 4 x^{2} - 4 x + \left (4 x + 2\right ) \log {\relax (x )} - 1}{\log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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