Integrand size = 18, antiderivative size = 33 \[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} (1-x)}{\sqrt {-2+4 x-3 x^2}}\right )}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {738, 210} \[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} (1-x)}{\sqrt {-3 x^2+4 x-2}}\right )}{\sqrt {2}} \]
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Rule 210
Rule 738
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=\frac {\arctan \left (\frac {-1+x}{\sqrt {-1+2 x-\frac {3 x^2}{2}}}\right )}{\sqrt {2}} \]
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 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 | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {-3 x^{2}+4 x -2}+2 x -2}{x}\right )}{2}\) | \(38\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (26) = 52\).
Time = 0.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=\frac {1}{4} \, \sqrt {-2} \log \left (\frac {\sqrt {-2} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 2 \, x - 2}{x}\right ) - \frac {1}{4} \, \sqrt {-2} \log \left (-\frac {\sqrt {-2} \sqrt {-3 \, x^{2} + 4 \, x - 2} - 2 \, x + 2}{x}\right ) \]
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\[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=\int \frac {1}{x \sqrt {- 3 x^{2} + 4 x - 2}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=\frac {1}{2} i \, \sqrt {2} \operatorname {arsinh}\left (\frac {\sqrt {2} x}{{\left | x \right |}} - \frac {\sqrt {2}}{{\left | x \right |}}\right ) \]
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\[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{2} + 4 \, x - 2} x} \,d x } \]
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Time = 10.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x \sqrt {-2+4 x-3 x^2}} \, dx=\frac {\sqrt {2}\,\ln \left (\frac {2\,x-2+\sqrt {2}\,\sqrt {-3\,x^2+4\,x-2}\,1{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \]
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