Optimal. Leaf size=40 \[ -\frac {1}{4} \sqrt {x^2+1} x+\frac {1}{2} x^2 \log \left (\sqrt {x^2+1}+x\right )+\frac {1}{4} \sinh ^{-1}(x) \]
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Rubi [A] time = 0.02, 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 = {2536, 321, 215} \[ -\frac {1}{4} \sqrt {x^2+1} x+\frac {1}{2} x^2 \log \left (\sqrt {x^2+1}+x\right )+\frac {1}{4} \sinh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 215
Rule 321
Rule 2536
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 \[ \frac {1}{4} \left (-\sqrt {x^2+1} x+2 x^2 \log \left (\sqrt {x^2+1}+x\right )+\sinh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 30, normalized size = 0.75 \[ \frac {1}{4} \, {\left (2 \, x^{2} + 1\right )} \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {x^{2} + 1} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 40, normalized size = 1.00 \[ \frac {1}{2} \, x^{2} \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {x^{2} + 1} x - \frac {1}{4} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, 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 \[ \frac {1}{2} \, x^{2} \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, x^{2} - \int \frac {x^{2}}{2 \, {\left (x^{3} + {\left (x^{2} + 1\right )}^{\frac {3}{2}} + x\right )}}\,{d x} + \frac {1}{4} \, \log \left (x^{2} + 1\right ) \]
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 \[ x\,\ln \left (x+\sqrt {x^2+1}\right )\,\left (\frac {x}{2}+\frac {1}{4\,x}\right )-\frac {x\,\sqrt {x^2+1}}{4} \]
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 \[ \int x \log {\left (x + \sqrt {x^{2} + 1} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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