Optimal. Leaf size=26 \[ \sqrt {1+x^2}+x \log \left (x-\sqrt {1+x^2}\right ) \]
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Rubi [A]
time = 0.00, 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 = {2614, 267}
\begin {gather*} \sqrt {x^2+1}+x \log \left (x-\sqrt {x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2614
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 \begin {gather*} \sqrt {1+x^2}+x \log \left (x-\sqrt {1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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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 \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 = 0.94, size = 22, normalized size = 0.85 \begin {gather*} x \log \left (x - \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.79, size = 20, normalized size = 0.77 \begin {gather*} x \log {\left (x - \sqrt {x^{2} + 1} \right )} + \sqrt {x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.55, size = 22, normalized size = 0.85 \begin {gather*} x \log \left (x - \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} \end {gather*}
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 \begin {gather*} x\,\ln \left (x-\sqrt {x^2+1}\right )+\sqrt {x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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