Optimal. Leaf size=26 \[ -x+\sqrt {-1+x^2} \log \left (x+\sqrt {-1+x^2}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {267, 2634, 8}
\begin {gather*} \sqrt {x^2-1} \log \left (\sqrt {x^2-1}+x\right )-x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 267
Rule 2634
Rubi steps
\begin {align*} \int \frac {x \log \left (x+\sqrt {-1+x^2}\right )}{\sqrt {-1+x^2}} \, dx &=\sqrt {-1+x^2} \log \left (x+\sqrt {-1+x^2}\right )-\int 1 \, dx\\ &=-x+\sqrt {-1+x^2} \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*} -x+\sqrt {-1+x^2} \log \left (x+\sqrt {-1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {x^{2}-1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 10.71, size = 22, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 22, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.05, size = 20, normalized size = 0.77 \begin {gather*} - x + \sqrt {x^{2} - 1} \log {\left (x + \sqrt {x^{2} - 1} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 22, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \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 = -0.04 \begin {gather*} \int \frac {x\,\ln \left (x+\sqrt {x^2-1}\right )}{\sqrt {x^2-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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