Optimal. Leaf size=33 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-2+4 x-3 x^2}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {738, 210}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-3 x^2+4 x-2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 738
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4+4 x}{\sqrt {-2+4 x-3 x^2}}\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-2+4 x-3 x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 43, normalized size = 1.30 \begin {gather*} -i \sqrt {2} \tanh ^{-1}\left (\sqrt {\frac {3}{2}} x+\frac {i \sqrt {-2+4 x-3 x^2}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 29, normalized size = 0.88
method | result | size |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\left (-4+4 x \right ) \sqrt {2}}{4 \sqrt {-3 x^{2}+4 x -2}}\right )}{2}\) | \(29\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \sqrt {-3 x^{2}+4 x -2}-2 x +2}{x}\right )}{2}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{2} i \, \sqrt {2} \operatorname {arsinh}\left (\frac {\sqrt {2} x}{{\left | x \right |}} - \frac {\sqrt {2}}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs.
\(2 (26) = 52\).
time = 2.72, size = 64, normalized size = 1.94 \begin {gather*} \frac {1}{4} \, \sqrt {-2} \log \left (\frac {\sqrt {-2} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 2 \, x - 2}{x}\right ) - \frac {1}{4} \, \sqrt {-2} \log \left (-\frac {\sqrt {-2} \sqrt {-3 \, x^{2} + 4 \, x - 2} - 2 \, x + 2}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- 3 x^{2} + 4 x - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 34, normalized size = 1.03 \begin {gather*} \frac {\sqrt {2}\,\ln \left (\frac {2\,x-2+\sqrt {2}\,\sqrt {-3\,x^2+4\,x-2}\,1{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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