Integrand size = 18, antiderivative size = 45 \[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\frac {\arctan \left (\left (-1+\sqrt {2}\right ) \sqrt {-3+x}\right )}{\sqrt {2}}+\frac {\arctan \left (\left (1+\sqrt {2}\right ) \sqrt {-3+x}\right )}{\sqrt {2}} \]
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Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {841, 1177, 209} \[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {x-3}}{\sqrt {3-2 \sqrt {2}}}\right )}{\sqrt {2}}+\frac {\arctan \left (\frac {\sqrt {x-3}}{\sqrt {3+2 \sqrt {2}}}\right )}{\sqrt {2}} \]
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Rule 209
Rule 841
Rule 1177
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1+x^2}{1+6 x^2+x^4} \, dx,x,\sqrt {-3+x}\right ) \\ & = \frac {1}{2} \left (2-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{3-2 \sqrt {2}+x^2} \, dx,x,\sqrt {-3+x}\right )+\frac {1}{2} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{3+2 \sqrt {2}+x^2} \, dx,x,\sqrt {-3+x}\right ) \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {-3+x}}{\sqrt {3-2 \sqrt {2}}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-3+x}}{\sqrt {3+2 \sqrt {2}}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.58 \[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\frac {\arctan \left (\frac {-4+x}{2 \sqrt {2} \sqrt {-3+x}}\right )}{\sqrt {2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.38
method | result | size |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+8 \sqrt {-3+x}\, x -16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x -32 \sqrt {-3+x}+40 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{x^{2}-8}\right )}{4}\) | \(62\) |
derivativedivides | \(\frac {\left (1+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \sqrt {-3+x}}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}+\frac {\left (\sqrt {2}-1\right ) \sqrt {2}\, \arctan \left (\frac {2 \sqrt {-3+x}}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}\) | \(72\) |
default | \(\frac {\left (1+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \sqrt {-3+x}}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}+\frac {\left (\sqrt {2}-1\right ) \sqrt {2}\, \arctan \left (\frac {2 \sqrt {-3+x}}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}\) | \(72\) |
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none
Time = 0.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.42 \[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 4\right )}}{4 \, \sqrt {x - 3}}\right ) \]
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\[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\int \frac {x - 2}{\sqrt {x - 3} \left (x^{2} - 8\right )}\, dx \]
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\[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\int { \frac {x - 2}{{\left (x^{2} - 8\right )} \sqrt {x - 3}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.51 \[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\frac {1}{4} \, \sqrt {2} {\left (\pi + 2 \, \arctan \left (\frac {\sqrt {2} {\left (x - 4\right )}}{4 \, \sqrt {x - 3}}\right )\right )} \]
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Time = 0.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx=\frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-3}}{4}\right )+\mathrm {atan}\left (\frac {7\,\sqrt {2}\,\sqrt {x-3}}{4}+\frac {\sqrt {2}\,{\left (x-3\right )}^{3/2}}{4}\right )\right )}{2} \]
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