Optimal. Leaf size=34 \[ -\sqrt {-1+x^2}+\tan ^{-1}\left (\sqrt {-1+x^2}\right )+\sqrt {-1+x^2} \log (x) \]
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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 = {2376, 272, 52,
65, 209} \begin {gather*} \text {ArcTan}\left (\sqrt {x^2-1}\right )-\sqrt {x^2-1}+\sqrt {x^2-1} \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 272
Rule 2376
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} \text {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} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^2\right )\\ &=-\sqrt {-1+x^2}+\sqrt {-1+x^2} \log (x)+\text {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 \begin {gather*} -\tan ^{-1}\left (\frac {1}{\sqrt {-1+x^2}}\right )+\sqrt {-1+x^2} (-1+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.05, size = 119, normalized size = 3.50
method | result | size |
meijerg | \(-\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \left (2-2 \sqrt {-x^{2}+1}\right )}{4 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \ln \left (x \right ) \left (2-2 \sqrt {-x^{2}+1}\right )}{2 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \left (-16+16 \sqrt {-x^{2}+1}-32 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{2}+1}}{2}\right )\right )}{32 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.58, size = 27, normalized size = 0.79 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x\right ) - \sqrt {x^{2} - 1} - \arcsin \left (\frac {1}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.22, size = 27, normalized size = 0.79 \begin {gather*} \sqrt {x^{2} - 1} {\left (\log \left (x\right ) - 1\right )} + 2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.43, size = 29, normalized size = 0.85 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.94, size = 28, normalized size = 0.82 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x\right ) - \sqrt {x^{2} - 1} + \arctan \left (\sqrt {x^{2} - 1}\right ) \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 = -0.03 \begin {gather*} \int \frac {x\,\ln \left (x\right )}{\sqrt {x^2-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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