Integrand size = 18, antiderivative size = 26 \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}+\frac {\text {arccosh}(x)}{2} \]
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Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {92, 54} \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {\text {arccosh}(x)}{2}+\frac {1}{2} \sqrt {x-1} \sqrt {x+1} x \]
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Rule 54
Rule 92
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}+\frac {1}{2} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx \\ & = \frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}+\frac {1}{2} \cosh ^{-1}(x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}+\text {arctanh}\left (\sqrt {\frac {-1+x}{1+x}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).
Time = 1.77 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54
method | result | size |
default | \(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \left (x \sqrt {x^{2}-1}+\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2 \sqrt {x^{2}-1}}\) | \(40\) |
risch | \(\frac {x \sqrt {-1+x}\, \sqrt {1+x}}{2}+\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (-1+x \right ) \left (1+x \right )}}{2 \sqrt {-1+x}\, \sqrt {1+x}}\) | \(46\) |
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none
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} x - \frac {1}{2} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \]
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Timed out. \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\text {Timed out} \]
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none
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {1}{2} \, \sqrt {x^{2} - 1} x + \frac {1}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} x - \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \]
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Time = 7.89 (sec) , antiderivative size = 194, normalized size of antiderivative = 7.46 \[ \int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx=2\,\mathrm {atanh}\left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )-\frac {\frac {14\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {14\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {x+1}-1\right )}^5}+\frac {2\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {x+1}-1\right )}^7}+\frac {2\,\left (\sqrt {x-1}-\mathrm {i}\right )}{\sqrt {x+1}-1}}{1+\frac {6\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}} \]
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