Integrand size = 25, antiderivative size = 17 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 = 0.20 (sec) , antiderivative size = 17, 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 = 2.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06
method | result | size |
elliptic | \(-\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\) | \(18\) |
trager | \(\frac {\ln \left (-\frac {-x^{4}+2 x \sqrt {x^{4}+x^{2}+1}-2 x^{2}-1}{x^{4}+1}\right )}{2}\) | \(38\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, x^{2}+\sqrt {2}-x}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {x +\sqrt {2}\, x^{2}+\sqrt {2}}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}\) | \(56\) |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, x^{2}+\sqrt {2}-x}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {x +\sqrt {2}\, x^{2}+\sqrt {2}}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}\) | \(56\) |
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {1}{2} \, \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + 1} x + 1}{x^{4} + 1}\right ) \]
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\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{4} + 1\right )}\, dx \]
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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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