Optimal. Leaf size=14 \[ x \left (1+\frac {3 x}{2}+\log (3+\log (x))\right ) \]
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Rubi [F] time = 0.12, 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+9 x+(1+3 x) \log (x)+(3+\log (x)) \log (3+\log (x))}{3+\log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4+9 x+\log (x)+3 x \log (x)}{3+\log (x)}+\log (3+\log (x))\right ) \, dx\\ &=\int \frac {4+9 x+\log (x)+3 x \log (x)}{3+\log (x)} \, dx+\int \log (3+\log (x)) \, dx\\ &=\int \left (1+3 x+\frac {1}{3+\log (x)}\right ) \, dx+\int \log (3+\log (x)) \, dx\\ &=x+\frac {3 x^2}{2}+\int \frac {1}{3+\log (x)} \, dx+\int \log (3+\log (x)) \, dx\\ &=x+\frac {3 x^2}{2}+\int \log (3+\log (x)) \, dx+\operatorname {Subst}\left (\int \frac {e^x}{3+x} \, dx,x,\log (x)\right )\\ &=x+\frac {3 x^2}{2}+\frac {\text {Ei}(3+\log (x))}{e^3}+\int \log (3+\log (x)) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 16, normalized size = 1.14 \begin {gather*} x+\frac {3 x^2}{2}+x \log (3+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{2} \, x^{2} + x \log \left (\log \relax (x) + 3\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{2} \, x^{2} + x \log \left (\log \relax (x) + 3\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 15, normalized size = 1.07
method | result | size |
norman | \(x +x \ln \left (3+\ln \relax (x )\right )+\frac {3 x^{2}}{2}\) | \(15\) |
risch | \(x +x \ln \left (3+\ln \relax (x )\right )+\frac {3 x^{2}}{2}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -e^{\left (-3\right )} E_{1}\left (-\log \relax (x) - 3\right ) \log \relax (x) - 3 \, e^{\left (-6\right )} E_{1}\left (-2 \, \log \relax (x) - 6\right ) \log \relax (x) + e^{\left (-3\right )} E_{2}\left (-\log \relax (x) - 3\right ) + \frac {3}{2} \, e^{\left (-6\right )} E_{2}\left (-2 \, \log \relax (x) - 6\right ) - 4 \, e^{\left (-3\right )} E_{1}\left (-\log \relax (x) - 3\right ) - 9 \, e^{\left (-6\right )} E_{1}\left (-2 \, \log \relax (x) - 6\right ) + x \log \left (\log \relax (x) + 3\right ) - \int \frac {1}{\log \relax (x) + 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 15, normalized size = 1.07 \begin {gather*} \frac {x\,\left (3\,x+2\,\ln \left (\ln \relax (x)+3\right )+2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 15, normalized size = 1.07 \begin {gather*} \frac {3 x^{2}}{2} + x \log {\left (\log {\relax (x )} + 3 \right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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