Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} (x+1)}{\sqrt {-3 x^2+4 x+2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 31, 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 = {724, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} (x+1)}{\sqrt {-3 x^2+4 x+2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {2+4 x-3 x^2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+4 x}{\sqrt {2+4 x-3 x^2}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} (1+x)}{\sqrt {2+4 x-3 x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.90 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x+1}{\sqrt {-\frac {3 x^2}{2}+2 x+1}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.13, size = 43, normalized size = 1.39 \begin {gather*} i \sqrt {2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x+\frac {i \sqrt {-3 x^2+4 x+2}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 39, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (x + 1\right )} + x^{2} - 8 \, x - 4}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 98, normalized size = 3.16 \begin {gather*} -\frac {1}{6} \, \sqrt {6} \sqrt {3} \log \left (\frac {{\left | -14 \, \sqrt {10} - 14 \, \sqrt {6} + \frac {28 \, {\left (\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} - \sqrt {10}\right )}}{3 \, x - 2} \right |}}{{\left | -14 \, \sqrt {10} + 14 \, \sqrt {6} + \frac {28 \, {\left (\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} - \sqrt {10}\right )}}{3 \, x - 2} \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 29, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4 x +4\right ) \sqrt {2}}{4 \sqrt {-3 x^{2}+4 x +2}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.90, size = 35, normalized size = 1.13 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {-3 \, x^{2} + 4 \, x + 2}}{{\left | x \right |}} + \frac {4}{{\left | x \right |}} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 27, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {2}\,\ln \left (\frac {2\,x+\sqrt {-6\,x^2+8\,x+4}+2}{x}\right )}{2} \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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