Optimal. Leaf size=26 \[ \sqrt {x^2+1}+x \log \left (x-\sqrt {x^2+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2534, 261} \[ \sqrt {x^2+1}+x \log \left (x-\sqrt {x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 261
Rule 2534
Rubi steps
\begin {align*} \int \log \left (x-\sqrt {1+x^2}\right ) \, dx &=x \log \left (x-\sqrt {1+x^2}\right )+\int \frac {x}{\sqrt {1+x^2}} \, dx\\ &=\sqrt {1+x^2}+x \log \left (x-\sqrt {1+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.00 \[ \sqrt {x^2+1}+x \log \left (x-\sqrt {x^2+1}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (x-\sqrt {1+x^2}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.06, size = 22, normalized size = 0.85 \[ x \log \left (x - \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 22, normalized size = 0.85 \[ x \log \left (x - \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.88
method | result | size |
default | \(x \ln \left (x -\sqrt {x^{2}+1}\right )+\sqrt {x^{2}+1}\) | \(23\) |
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 \[ x \log \left (x - \sqrt {x^{2} + 1}\right ) - x + \arctan \relax (x) + \int -\frac {x}{x^{3} - {\left (x^{2} + 1\right )}^{\frac {3}{2}} + x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 22, normalized size = 0.85 \[ x\,\ln \left (x-\sqrt {x^2+1}\right )+\sqrt {x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.26, size = 20, normalized size = 0.77 \[ x \log {\left (x - \sqrt {x^{2} + 1} \right )} + \sqrt {x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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