Optimal. Leaf size=42 \[ \frac {1}{8} \sqrt {\log ^2(x)+1} \log (x)+\frac {1}{4} \sqrt {\log ^2(x)+1} \log ^3(x)-\frac {1}{8} \sinh ^{-1}(\log (x)) \]
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Rubi [A] time = 0.07, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {279, 321, 215} \[ \frac {1}{4} \sqrt {\log ^2(x)+1} \log ^3(x)+\frac {1}{8} \sqrt {\log ^2(x)+1} \log (x)-\frac {1}{8} \sinh ^{-1}(\log (x)) \]
Antiderivative was successfully verified.
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Rule 215
Rule 279
Rule 321
Rubi steps
\begin {align*} \int \frac {\log ^2(x) \sqrt {1+\log ^2(x)}}{x} \, dx &=\operatorname {Subst}\left (\int x^2 \sqrt {1+x^2} \, dx,x,\log (x)\right )\\ &=\frac {1}{4} \log ^3(x) \sqrt {1+\log ^2(x)}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,\log (x)\right )\\ &=\frac {1}{8} \log (x) \sqrt {1+\log ^2(x)}+\frac {1}{4} \log ^3(x) \sqrt {1+\log ^2(x)}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\log (x)\right )\\ &=-\frac {1}{8} \sinh ^{-1}(\log (x))+\frac {1}{8} \log (x) \sqrt {1+\log ^2(x)}+\frac {1}{4} \log ^3(x) \sqrt {1+\log ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.74 \[ \frac {1}{8} \left (\log (x) \sqrt {\log ^2(x)+1} \left (2 \log ^2(x)+1\right )-\sinh ^{-1}(\log (x))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 36, normalized size = 0.86 \[ \frac {1}{8} \, {\left (2 \, \log \relax (x)^{3} + \log \relax (x)\right )} \sqrt {\log \relax (x)^{2} + 1} + \frac {1}{8} \, \log \left (\sqrt {\log \relax (x)^{2} + 1} - \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 37, normalized size = 0.88 \[ \frac {1}{8} \, {\left (2 \, \log \relax (x)^{2} + 1\right )} \sqrt {\log \relax (x)^{2} + 1} \log \relax (x) + \frac {1}{8} \, \log \left (\sqrt {\log \relax (x)^{2} + 1} - \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 31, normalized size = 0.74 \[ -\frac {\arcsinh \left (\ln \relax (x )\right )}{8}+\frac {\left (\ln \relax (x )^{2}+1\right )^{\frac {3}{2}} \ln \relax (x )}{4}-\frac {\sqrt {\ln \relax (x )^{2}+1}\, \ln \relax (x )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 30, normalized size = 0.71 \[ \frac {1}{4} \, {\left (\log \relax (x)^{2} + 1\right )}^{\frac {3}{2}} \log \relax (x) - \frac {1}{8} \, \sqrt {\log \relax (x)^{2} + 1} \log \relax (x) - \frac {1}{8} \, \operatorname {arsinh}\left (\log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 26, normalized size = 0.62 \[ \left (\frac {{\ln \relax (x)}^3}{4}+\frac {\ln \relax (x)}{8}\right )\,\sqrt {{\ln \relax (x)}^2+1}-\frac {\mathrm {asinh}\left (\ln \relax (x)\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\log {\relax (x )}^{2} + 1} \log {\relax (x )}^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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