Optimal. Leaf size=40 \[ -\frac {1}{4} x \sqrt {1+x^2}+\frac {1}{4} \sinh ^{-1}(x)+\frac {1}{2} x^2 \log \left (x+\sqrt {1+x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2616, 327, 221}
\begin {gather*} -\frac {1}{4} \sqrt {x^2+1} x+\frac {1}{2} x^2 \log \left (\sqrt {x^2+1}+x\right )+\frac {1}{4} \sinh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 2616
Rubi steps
\begin {align*} \int x \log \left (x+\sqrt {1+x^2}\right ) \, dx &=\frac {1}{2} x^2 \log \left (x+\sqrt {1+x^2}\right )-\frac {1}{2} \int \frac {x^2}{\sqrt {1+x^2}} \, dx\\ &=-\frac {1}{4} x \sqrt {1+x^2}+\frac {1}{2} x^2 \log \left (x+\sqrt {1+x^2}\right )+\frac {1}{4} \int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=-\frac {1}{4} x \sqrt {1+x^2}+\frac {1}{4} \sinh ^{-1}(x)+\frac {1}{2} x^2 \log \left (x+\sqrt {1+x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.90 \begin {gather*} \frac {1}{4} \left (-x \sqrt {1+x^2}+\sinh ^{-1}(x)+2 x^2 \log \left (x+\sqrt {1+x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x \ln \left (x +\sqrt {x^{2}+1}\right )\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 30, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, {\left (2 \, x^{2} + 1\right )} \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {x^{2} + 1} x \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 x \log {\left (x + \sqrt {x^{2} + 1} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 51, normalized size = 1.28 \begin {gather*} -\frac {2}{8} x \sqrt {x^{2}+1}-\frac {\ln \left (\sqrt {x^{2}+1}-x\right )}{4}+\frac {1}{2} x^{2} \ln \left (x+\sqrt {x^{2}+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 32, normalized size = 0.80 \begin {gather*} x\,\ln \left (x+\sqrt {x^2+1}\right )\,\left (\frac {x}{2}+\frac {1}{4\,x}\right )-\frac {x\,\sqrt {x^2+1}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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