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 [A]
time = 0.05, size = 39, normalized size = 1.18 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x-\frac {\sqrt {-2+4 x+3 x^2}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, 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 | \(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \sqrt {3 x^{2}+4 x -2}+2 x -2}{x}\right )}{2}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 26, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arcsin \left (\frac {\sqrt {10} x}{5 \, {\left | x \right |}} - \frac {\sqrt {10}}{5 \, {\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.84, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )}}{\sqrt {3 \, x^{2} + 4 \, x - 2}}\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 [A]
time = 1.55, size = 30, normalized size = 0.91 \begin {gather*} \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x - 2}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, 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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