Integrand size = 24, antiderivative size = 69 \[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=\frac {1}{4} \sqrt {-x^2+x^4}+\text {arctanh}\left (\frac {(-1+x) x}{\sqrt {-x^2+x^4}}\right )-\frac {1}{4} \sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {-x^2+x^4}}{x^2}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2059, 748, 857, 634, 212, 738} \[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=\frac {1}{2} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4-x^2}}\right )+\frac {1}{8} \sqrt {3} \text {arctanh}\left (\frac {3-4 x^2}{2 \sqrt {3} \sqrt {x^4-x^2}}\right )+\frac {1}{4} \sqrt {x^4-x^2} \]
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Rule 212
Rule 634
Rule 738
Rule 748
Rule 857
Rule 2059
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-x+x^2}}{-3+2 x} \, dx,x,x^2\right ) \\ & = \frac {1}{4} \sqrt {-x^2+x^4}-\frac {1}{8} \text {Subst}\left (\int \frac {3-4 x}{(-3+2 x) \sqrt {-x+x^2}} \, dx,x,x^2\right ) \\ & = \frac {1}{4} \sqrt {-x^2+x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-x+x^2}} \, dx,x,x^2\right )+\frac {3}{8} \text {Subst}\left (\int \frac {1}{(-3+2 x) \sqrt {-x+x^2}} \, dx,x,x^2\right ) \\ & = \frac {1}{4} \sqrt {-x^2+x^4}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-x^2+x^4}}\right )-\frac {3}{4} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {-3+4 x^2}{\sqrt {-x^2+x^4}}\right ) \\ & = \frac {1}{4} \sqrt {-x^2+x^4}+\frac {1}{2} \text {arctanh}\left (\frac {x^2}{\sqrt {-x^2+x^4}}\right )+\frac {1}{8} \sqrt {3} \text {arctanh}\left (\frac {3-4 x^2}{2 \sqrt {3} \sqrt {-x^2+x^4}}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.17 \[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=\frac {x \sqrt {-1+x^2} \left (x \sqrt {-1+x^2}-\sqrt {3} \text {arctanh}\left (\frac {x}{\sqrt {3} \sqrt {-1+x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt {-1+x^2}}{-1+x}\right )\right )}{4 \sqrt {x^2 \left (-1+x^2\right )}} \]
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Time = 1.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {\sqrt {x^{4}-x^{2}}}{4}+\frac {\ln \left (2 x^{2}-1+2 \sqrt {x^{4}-x^{2}}\right )}{4}-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (4 x^{2}-3\right ) \sqrt {3}}{6 \sqrt {x^{4}-x^{2}}}\right )}{8}\) | \(67\) |
trager | \(\frac {\sqrt {x^{4}-x^{2}}}{4}-\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x^{2}}}{x}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}-x^{2}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{2 x^{2}-3}\right )}{8}\) | \(91\) |
default | \(-\frac {\sqrt {x^{4}-x^{2}}\, \left (\sqrt {6}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\sqrt {6}\, x +2\right ) \sqrt {2}}{2 \sqrt {x^{2}-1}}\right )+\sqrt {6}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\sqrt {6}\, x -2\right ) \sqrt {2}}{2 \sqrt {x^{2}-1}}\right )-4 x \sqrt {x^{2}-1}-8 \ln \left (x +\sqrt {x^{2}-1}\right )\right )}{16 x \sqrt {x^{2}-1}}\) | \(101\) |
risch | \(\frac {\sqrt {x^{2} \left (x^{2}-1\right )}}{4}+\frac {\left (\frac {\ln \left (x +\sqrt {x^{2}-1}\right )}{2}+\frac {\sqrt {6}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1-\sqrt {6}\, \left (x +\frac {\sqrt {6}}{2}\right )\right ) \sqrt {2}}{\sqrt {4 \left (x +\frac {\sqrt {6}}{2}\right )^{2}-4 \sqrt {6}\, \left (x +\frac {\sqrt {6}}{2}\right )+2}}\right )}{16}-\frac {\sqrt {6}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1+\sqrt {6}\, \left (x -\frac {\sqrt {6}}{2}\right )\right ) \sqrt {2}}{\sqrt {4 \left (x -\frac {\sqrt {6}}{2}\right )^{2}+4 \sqrt {6}\, \left (x -\frac {\sqrt {6}}{2}\right )+2}}\right )}{16}\right ) \sqrt {x^{2} \left (x^{2}-1\right )}}{x \sqrt {x^{2}-1}}\) | \(157\) |
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none
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.36 \[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=\frac {1}{8} \, \sqrt {3} \log \left (\frac {8 \, x^{2} - \sqrt {3} {\left (4 \, x^{2} - 3\right )} - 2 \, \sqrt {x^{4} - x^{2}} {\left (2 \, \sqrt {3} - 3\right )} - 6}{2 \, x^{2} - 3}\right ) + \frac {1}{4} \, \sqrt {x^{4} - x^{2}} - \frac {1}{2} \, \log \left (-\frac {x^{2} - \sqrt {x^{4} - x^{2}}}{x}\right ) \]
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\[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=\int \frac {x \sqrt {x^{2} \left (x - 1\right ) \left (x + 1\right )}}{2 x^{2} - 3}\, dx \]
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\[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=\int { \frac {\sqrt {x^{4} - x^{2}} x}{2 \, x^{2} - 3} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.72 \[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=-\frac {1}{24} \, \sqrt {3} {\left (-2 i \, \sqrt {3} \pi - 3 \, \log \left (-\frac {\sqrt {3} + 3}{\sqrt {3} - 3}\right )\right )} \mathrm {sgn}\left (x\right ) + \frac {1}{4} \, \sqrt {x^{2} - 1} x \mathrm {sgn}\left (x\right ) - \frac {1}{8} \, \sqrt {3} \log \left (\frac {{\left | 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 2 \, \sqrt {3} - 4 \right |}}{{\left | 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, \sqrt {3} - 4 \right |}}\right ) \mathrm {sgn}\left (x\right ) - \frac {1}{4} \, \log \left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) \]
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Timed out. \[ \int \frac {x \sqrt {-x^2+x^4}}{-3+2 x^2} \, dx=\int \frac {x\,\sqrt {x^4-x^2}}{2\,x^2-3} \,d x \]
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