Integrand size = 27, antiderivative size = 15 \[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 15, 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 = {2137, 212} \[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {x^4+x^2-1}}\right ) \]
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Rule 212
Rule 2137
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \\ & = \text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \]
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Time = 3.76 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
elliptic | \(\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )\) | \(16\) |
trager | \(\frac {\ln \left (-\frac {x^{4}+2 x \sqrt {x^{4}+x^{2}-1}+2 x^{2}-1}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\) | \(46\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}\) | \(62\) |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27 \[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\frac {1}{2} \, \log \left (\frac {x^{4} + 2 \, x^{2} + 2 \, \sqrt {x^{4} + x^{2} - 1} x - 1}{x^{4} - 1}\right ) \]
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\[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=- \int \frac {x^{4}}{x^{4} \sqrt {x^{4} + x^{2} - 1} - \sqrt {x^{4} + x^{2} - 1}}\, dx - \int \frac {1}{x^{4} \sqrt {x^{4} + x^{2} - 1} - \sqrt {x^{4} + x^{2} - 1}}\, dx \]
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\[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\int { -\frac {x^{4} + 1}{\sqrt {x^{4} + x^{2} - 1} {\left (x^{4} - 1\right )}} \,d x } \]
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\[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\int { -\frac {x^{4} + 1}{\sqrt {x^{4} + x^{2} - 1} {\left (x^{4} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\int -\frac {x^4+1}{\left (x^4-1\right )\,\sqrt {x^4+x^2-1}} \,d x \]
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