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 = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 203
Rule 266
Rule 2338
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.44, size = 27, normalized size = 0.79 \[ \sqrt {x^{2} - 1} {\left (\log \relax (x) - 1\right )} + 2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 28, normalized size = 0.82 \[ \sqrt {x^{2} - 1} \log \relax (x) - \sqrt {x^{2} - 1} + \arctan \left (\sqrt {x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 119, normalized size = 3.50 \[ \frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \left (2-2 \sqrt {-x^{2}+1}\right ) \ln \relax (x )}{2 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \left (2-2 \sqrt {-x^{2}+1}\right )}{4 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \left (-32 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{2}+1}}{2}\right )-16+16 \sqrt {-x^{2}+1}\right )}{32 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 27, normalized size = 0.79 \[ \sqrt {x^{2} - 1} \log \relax (x) - \sqrt {x^{2} - 1} - \arcsin \left (\frac {1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\ln \relax (x)}{\sqrt {x^2-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.74, size = 29, normalized size = 0.85 \[ \sqrt {x^{2} - 1} \log {\relax (x )} - \begin {cases} \sqrt {x^{2} - 1} - \operatorname {acos}{\left (\frac {1}{x} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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