Optimal. Leaf size=34 \[ -\sqrt {1+x^2}+\tanh ^{-1}\left (\sqrt {1+x^2}\right )+\sqrt {1+x^2} \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2376, 272, 52,
65, 213} \begin {gather*} -\sqrt {x^2+1}+\sqrt {x^2+1} \log (x)+\tanh ^{-1}\left (\sqrt {x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 213
Rule 272
Rule 2376
Rubi steps
\begin {align*} \int \frac {x \log (x)}{\sqrt {1+x^2}} \, dx &=\sqrt {1+x^2} \log (x)-\int \frac {\sqrt {1+x^2}}{x} \, dx\\ &=\sqrt {1+x^2} \log (x)-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^2\right )\\ &=-\sqrt {1+x^2}+\sqrt {1+x^2} \log (x)-\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^2\right )\\ &=-\sqrt {1+x^2}+\sqrt {1+x^2} \log (x)-\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^2}\right )\\ &=-\sqrt {1+x^2}+\tanh ^{-1}\left (\sqrt {1+x^2}\right )+\sqrt {1+x^2} \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.18 \begin {gather*} -\sqrt {1+x^2}-\log (x)+\sqrt {1+x^2} \log (x)+\log \left (1+\sqrt {1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 39, normalized size = 1.15
method | result | size |
meijerg | \(1-\sqrt {x^{2}+1}+\frac {\ln \left (x \right ) \left (-2+2 \sqrt {x^{2}+1}\right )}{2}+\ln \left (\frac {1}{2}+\frac {\sqrt {x^{2}+1}}{2}\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.27, size = 25, normalized size = 0.74 \begin {gather*} \sqrt {x^{2} + 1} \log \left (x\right ) - \sqrt {x^{2} + 1} + \operatorname {arsinh}\left (\frac {1}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 41, normalized size = 1.21 \begin {gather*} \sqrt {x^{2} + 1} {\left (\log \left (x\right ) - 1\right )} + \log \left (-x + \sqrt {x^{2} + 1} + 1\right ) - \log \left (-x + \sqrt {x^{2} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.31, size = 41, normalized size = 1.21 \begin {gather*} - \frac {x}{\sqrt {1 + \frac {1}{x^{2}}}} + \sqrt {x^{2} + 1} \log {\left (x \right )} + \operatorname {asinh}{\left (\frac {1}{x} \right )} - \frac {1}{x \sqrt {1 + \frac {1}{x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 44, normalized size = 1.29 \begin {gather*} \sqrt {x^{2} + 1} \log \left (x\right ) - \sqrt {x^{2} + 1} + \frac {1}{2} \, \log \left (\sqrt {x^{2} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {x^{2} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x\,\ln \left (x\right )}{\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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