Optimal. Leaf size=42 \[ -\frac {1}{8} \sinh ^{-1}(\log (x))+\frac {1}{8} \log (x) \sqrt {1+\log ^2(x)}+\frac {1}{4} \log ^3(x) \sqrt {1+\log ^2(x)} \]
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Rubi [A]
time = 0.04, 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 = {285, 327, 221}
\begin {gather*} \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)) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 285
Rule 327
Rubi steps
\begin {align*} \int \frac {\log ^2(x) \sqrt {1+\log ^2(x)}}{x} \, dx &=\text {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} \text {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} \text {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 \begin {gather*} \frac {1}{8} \left (-\sinh ^{-1}(\log (x))+\log (x) \sqrt {1+\log ^2(x)} \left (1+2 \log ^2(x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 31, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {\ln \left (x \right ) \left (1+\ln \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {\ln \left (x \right ) \sqrt {1+\ln \left (x \right )^{2}}}{8}-\frac {\arcsinh \left (\ln \left (x \right )\right )}{8}\) | \(31\) |
default | \(\frac {\ln \left (x \right ) \left (1+\ln \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {\ln \left (x \right ) \sqrt {1+\ln \left (x \right )^{2}}}{8}-\frac {\arcsinh \left (\ln \left (x \right )\right )}{8}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 30, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, {\left (\log \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \log \left (x\right ) - \frac {1}{8} \, \sqrt {\log \left (x\right )^{2} + 1} \log \left (x\right ) - \frac {1}{8} \, \operatorname {arsinh}\left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{8} \, {\left (2 \, \log \left (x\right )^{3} + \log \left (x\right )\right )} \sqrt {\log \left (x\right )^{2} + 1} + \frac {1}{8} \, \log \left (\sqrt {\log \left (x\right )^{2} + 1} - \log \left (x\right )\right ) \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {\log {\left (x \right )}^{2} + 1} \log {\left (x \right )}^{2}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.92, size = 37, normalized size = 0.88 \begin {gather*} \frac {1}{8} \, {\left (2 \, \log \left (x\right )^{2} + 1\right )} \sqrt {\log \left (x\right )^{2} + 1} \log \left (x\right ) + \frac {1}{8} \, \log \left (\sqrt {\log \left (x\right )^{2} + 1} - \log \left (x\right )\right ) \end {gather*}
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 \begin {gather*} \left (\frac {{\ln \left (x\right )}^3}{4}+\frac {\ln \left (x\right )}{8}\right )\,\sqrt {{\ln \left (x\right )}^2+1}-\frac {\mathrm {asinh}\left (\ln \left (x\right )\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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