Optimal. Leaf size=26 \[ x \log \left (\sqrt{x^2-1}+x\right )-\sqrt{x^2-1} \]
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Rubi [A] time = 0.0044315, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2534, 261} \[ x \log \left (\sqrt{x^2-1}+x\right )-\sqrt{x^2-1} \]
Antiderivative was successfully verified.
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Rule 2534
Rule 261
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*}
Mathematica [A] time = 0.0164519, size = 26, normalized size = 1. \[ x \log \left (\sqrt{x^2-1}+x\right )-\sqrt{x^2-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 23, normalized size = 0.9 \begin{align*} x\ln \left ( x+\sqrt{{x}^{2}-1} \right ) -\sqrt{{x}^{2}-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} x \log \left (\sqrt{x + 1} \sqrt{x - 1} + x\right ) - x + \int \frac{x}{x^{3} +{\left (x^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right )\right )} - x}\,{d x} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9504, size = 57, normalized size = 2.19 \begin{align*} x \log \left (x + \sqrt{x^{2} - 1}\right ) - \sqrt{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.58059, size = 20, normalized size = 0.77 \begin{align*} x \log{\left (x + \sqrt{x^{2} - 1} \right )} - \sqrt{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23056, size = 30, normalized size = 1.15 \begin{align*} x \log \left (x + \sqrt{x^{2} - 1}\right ) - \sqrt{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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