Integrand size = 17, antiderivative size = 25 \[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=\sqrt {1-x^2} \sqrt {\frac {1}{-1+x^2}} \arcsin (x) \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6820, 1973, 222} \[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=\sqrt {1-x^2} \sqrt {\frac {1}{x^2-1}} \arcsin (x) \]
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Rule 222
Rule 1973
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\frac {1}{-1+x^2}} \, dx \\ & = \left (\sqrt {1-x^2} \sqrt {\frac {1}{-1+x^2}}\right ) \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \sqrt {1-x^2} \sqrt {\frac {1}{-1+x^2}} \sin ^{-1}(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(25)=50\).
Time = 0.00 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=\frac {1}{2} \sqrt {\frac {1}{-1+x^2}} \sqrt {-1+x^2} \left (-\log \left (1-\frac {x}{\sqrt {-1+x^2}}\right )+\log \left (1+\frac {x}{\sqrt {-1+x^2}}\right )\right ) \]
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Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
method | result | size |
default | \(\sqrt {\frac {1}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right )\) | \(28\) |
trager | \(\ln \left (\sqrt {\frac {1}{x^{2}-1}}\, x^{2}-\sqrt {\frac {1}{x^{2}-1}}+x \right )\) | \(28\) |
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none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=-\log \left (-x + \sqrt {x^{2} - 1}\right ) \]
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\[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=\int \sqrt {\frac {x^{2} + 1}{x^{4} - 1}}\, dx \]
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\[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=\int { \sqrt {\frac {x^{2} + 1}{x^{4} - 1}} \,d x } \]
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none
Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=-\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (x^{2} - 1\right ) \]
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Timed out. \[ \int \sqrt {\frac {1+x^2}{-1+x^4}} \, dx=\int \sqrt {\frac {x^2+1}{x^4-1}} \,d x \]
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