Integrand size = 27, antiderivative size = 26 \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1712, 212} \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {x^4+x^2+1}}\right )}{\sqrt {3}} \]
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Rule 212
Rule 1712
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}} \]
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Time = 1.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19
method | result | size |
elliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}\, \sqrt {6}}{6 x}\right ) \sqrt {6}\, \sqrt {2}}{6}\) | \(31\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\sqrt {x^{4}+x^{2}+1}}{\left (-1+x \right ) \left (1+x \right )}\right )}{3}\) | \(42\) |
default | \(\frac {\sqrt {3}\, \left (\operatorname {arctanh}\left (\frac {\left (2 x^{2}-x +2\right ) \sqrt {3}}{3 \sqrt {x^{4}+x^{2}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (2 x^{2}+x +2\right ) \sqrt {3}}{3 \sqrt {x^{4}+x^{2}+1}}\right )\right )}{6}\) | \(59\) |
pseudoelliptic | \(\frac {\sqrt {3}\, \left (\operatorname {arctanh}\left (\frac {\left (2 x^{2}-x +2\right ) \sqrt {3}}{3 \sqrt {x^{4}+x^{2}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (2 x^{2}+x +2\right ) \sqrt {3}}{3 \sqrt {x^{4}+x^{2}+1}}\right )\right )}{6}\) | \(59\) |
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{4} + 2 \, \sqrt {3} \sqrt {x^{4} + x^{2} + 1} x + 4 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) \]
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\[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=- \int \frac {x^{2}}{x^{2} \sqrt {x^{4} + x^{2} + 1} - \sqrt {x^{4} + x^{2} + 1}}\, dx - \int \frac {1}{x^{2} \sqrt {x^{4} + x^{2} + 1} - \sqrt {x^{4} + x^{2} + 1}}\, dx \]
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\[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { -\frac {x^{2} + 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}} \,d x } \]
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\[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { -\frac {x^{2} + 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int -\frac {x^2+1}{\left (x^2-1\right )\,\sqrt {x^4+x^2+1}} \,d x \]
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