Optimal. Leaf size=34 \[ -\sqrt{x^2-1}+\sqrt{x^2-1} \log (x)+\tan ^{-1}\left (\sqrt{x^2-1}\right ) \]
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Rubi [A] time = 0.0365319, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2338, 266, 50, 63, 203} \[ -\sqrt{x^2-1}+\sqrt{x^2-1} \log (x)+\tan ^{-1}\left (\sqrt{x^2-1}\right ) \]
Antiderivative was successfully verified.
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Rule 2338
Rule 266
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{x \log (x)}{\sqrt{-1+x^2}} \, dx &=\sqrt{-1+x^2} \log (x)-\int \frac{\sqrt{-1+x^2}}{x} \, dx\\ &=\sqrt{-1+x^2} \log (x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{-1+x}}{x} \, dx,x,x^2\right )\\ &=-\sqrt{-1+x^2}+\sqrt{-1+x^2} \log (x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x} \, dx,x,x^2\right )\\ &=-\sqrt{-1+x^2}+\sqrt{-1+x^2} \log (x)+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x^2}\right )\\ &=-\sqrt{-1+x^2}+\tan ^{-1}\left (\sqrt{-1+x^2}\right )+\sqrt{-1+x^2} \log (x)\\ \end{align*}
Mathematica [A] time = 0.0203238, size = 27, normalized size = 0.79 \[ \sqrt{x^2-1} (\log (x)-1)-\tan ^{-1}\left (\frac{1}{\sqrt{x^2-1}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0., size = 119, normalized size = 3.5 \begin{align*} -{\frac{1}{4}\sqrt{-{\it signum} \left ({x}^{2}-1 \right ) } \left ( 2-2\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{{\it signum} \left ({x}^{2}-1 \right ) }}}}+{\frac{\ln \left ( x \right ) }{2}\sqrt{-{\it signum} \left ({x}^{2}-1 \right ) } \left ( 2-2\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{{\it signum} \left ({x}^{2}-1 \right ) }}}}+{\frac{1}{32}\sqrt{-{\it signum} \left ({x}^{2}-1 \right ) } \left ( -16+16\,\sqrt{-{x}^{2}+1}-32\,\ln \left ( 1/2+1/2\,\sqrt{-{x}^{2}+1} \right ) \right ){\frac{1}{\sqrt{{\it signum} \left ({x}^{2}-1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41712, size = 36, normalized size = 1.06 \begin{align*} \sqrt{x^{2} - 1} \log \left (x\right ) - \sqrt{x^{2} - 1} - \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35498, size = 80, normalized size = 2.35 \begin{align*} \sqrt{x^{2} - 1}{\left (\log \left (x\right ) - 1\right )} + 2 \, \arctan \left (-x + \sqrt{x^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.89017, size = 29, normalized size = 0.85 \begin{align*} \sqrt{x^{2} - 1} \log{\left (x \right )} - \begin{cases} \sqrt{x^{2} - 1} - \operatorname{acos}{\left (\frac{1}{x} \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06846, size = 38, normalized size = 1.12 \begin{align*} \sqrt{x^{2} - 1} \log \left (x\right ) - \sqrt{x^{2} - 1} + \arctan \left (\sqrt{x^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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