Integrand size = 19, antiderivative size = 30 \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=\frac {\sqrt {-1+x^2} \sqrt {1+x^2} \text {arcsinh}(x)}{\sqrt {-1+x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1166, 221} \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=\frac {\sqrt {x^2-1} \sqrt {x^2+1} \text {arcsinh}(x)}{\sqrt {x^4-1}} \]
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Rule 221
Rule 1166
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1+x^2}} \, dx}{\sqrt {-1+x^4}} \\ & = \frac {\sqrt {-1+x^2} \sqrt {1+x^2} \sinh ^{-1}(x)}{\sqrt {-1+x^4}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=-\log \left (1-x^2\right )+\log \left (-x+x^3+\sqrt {-1+x^2} \sqrt {-1+x^4}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {x^{4}-1}\, \operatorname {arcsinh}\left (x \right )}{\sqrt {x^{2}-1}\, \sqrt {x^{2}+1}}\) | \(25\) |
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=\frac {1}{2} \, \log \left (\frac {x^{3} + \sqrt {x^{4} - 1} \sqrt {x^{2} - 1} - x}{x^{3} - x}\right ) - \frac {1}{2} \, \log \left (-\frac {x^{3} - \sqrt {x^{4} - 1} \sqrt {x^{2} - 1} - x}{x^{3} - x}\right ) \]
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\[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right )}}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} - 1}}{\sqrt {x^{4} - 1}} \,d x } \]
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\[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} - 1}}{\sqrt {x^{4} - 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1+x^4}} \, dx=\int \frac {\sqrt {x^2-1}}{\sqrt {x^4-1}} \,d x \]
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