Integrand size = 16, antiderivative size = 32 \[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {4+x}{2 \sqrt {2} \sqrt {2+x-x^2}}\right )}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {738, 212} \[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {x+4}{2 \sqrt {2} \sqrt {-x^2+x+2}}\right )}{\sqrt {2}} \]
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Rule 212
Rule 738
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+x}{\sqrt {2+x-x^2}}\right )\right ) \\ & = -\frac {\text {arctanh}\left (\frac {4+x}{2 \sqrt {2} \sqrt {2+x-x^2}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {2+x-x^2}}{-2+x}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\frac {\left (4+x \right ) \sqrt {2}}{4 \sqrt {-x^{2}+x +2}}\right ) \sqrt {2}}{2}\) | \(25\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {-x^{2}+x +2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x}\right )}{2}\) | \(43\) |
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none
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (-\frac {4 \, \sqrt {2} \sqrt {-x^{2} + x + 2} {\left (x + 4\right )} + 7 \, x^{2} - 16 \, x - 32}{x^{2}}\right ) \]
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\[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=\int \frac {1}{x \sqrt {- \left (x - 2\right ) \left (x + 1\right )}}\, dx \]
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none
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {-x^{2} + x + 2}}{{\left | x \right |}} + \frac {4}{{\left | x \right |}} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).
Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + \frac {2 \, {\left (2 \, \sqrt {-x^{2} + x + 2} - 3\right )}}{2 \, x - 1} - 6 \right |}}{{\left | 4 \, \sqrt {2} + \frac {2 \, {\left (2 \, \sqrt {-x^{2} + x + 2} - 3\right )}}{2 \, x - 1} - 6 \right |}}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \sqrt {2+x-x^2}} \, dx=-\frac {\sqrt {2}\,\ln \left (\frac {x+2\,\sqrt {2}\,\sqrt {-x^2+x+2}+4}{x}\right )}{2} \]
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