Integrand size = 22, antiderivative size = 24 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1713, 213} \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \]
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Rule 213
Rule 1713
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
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Time = 1.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
method | result | size |
elliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right ) \sqrt {2}}{2}\) | \(22\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}}{\left (x -1\right ) \left (1+x \right )}\right )}{2}\) | \(39\) |
default | \(\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right )}{4}\) | \(47\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right )}{4}\) | \(47\) |
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) \]
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\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\int \frac {x^{2} + 1}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1}}\, dx \]
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\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\int { \frac {x^{2} + 1}{\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}} \,d x } \]
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\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\int { \frac {x^{2} + 1}{\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\int \frac {x^2+1}{\left (x^2-1\right )\,\sqrt {x^4+1}} \,d x \]
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