Integrand size = 18, antiderivative size = 44 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=\frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \arctan \left (\sqrt {-1+\frac {1}{x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {25, 272, 43, 52, 65, 209} \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=-\frac {3}{2} \arctan \left (\sqrt {\frac {1}{x^2}-1}\right )-\frac {1}{2} \left (\frac {1}{x^2}-1\right )^{3/2} x^2+\frac {3}{2} \sqrt {\frac {1}{x^2}-1} \]
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Rule 25
Rule 43
Rule 52
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\int \left (-1+\frac {1}{x^2}\right )^{3/2} x \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(-1+x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {3}{4} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,\frac {1}{x^2}\right ) \\ & = \frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+\frac {1}{x^2}}\right ) \\ & = \frac {3}{2} \sqrt {-1+\frac {1}{x^2}}-\frac {1}{2} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {3}{2} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=\frac {1}{2} \sqrt {-1+\frac {1}{x^2}} \left (2+x^2-\frac {6 x \text {arctanh}\left (\frac {\sqrt {-1+x^2}}{-1+x}\right )}{\sqrt {-1+x^2}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.20
method | result | size |
trager | \(2 \left (\frac {x^{2}}{4}+\frac {1}{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\left (\sqrt {-\frac {x^{2}-1}{x^{2}}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right ) x \right )}{2}\) | \(53\) |
default | \(\frac {\sqrt {-\frac {x^{2}-1}{x^{2}}}\, \left (2 \left (-x^{2}+1\right )^{\frac {3}{2}}+3 x^{2} \sqrt {-x^{2}+1}+3 \arcsin \left (x \right ) x \right )}{2 \sqrt {-x^{2}+1}}\) | \(55\) |
risch | \(\frac {\left (x^{4}+x^{2}-2\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{2 x^{2}-2}-\frac {3 \arcsin \left (x \right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}\, x \sqrt {-x^{2}+1}}{2 \left (x^{2}-1\right )}\) | \(64\) |
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=\frac {1}{2} \, {\left (x^{2} + 2\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 3 \, \arctan \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) \]
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Time = 10.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=\frac {x^{2} \sqrt {-1 + \frac {1}{x^{2}}}}{2} + \sqrt {-1 + \frac {1}{x^{2}}} - \frac {3 \operatorname {atan}{\left (\sqrt {-1 + \frac {1}{x^{2}}} \right )}}{2} \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=\frac {1}{2} \, x^{2} \sqrt {\frac {1}{x^{2}} - 1} + \sqrt {\frac {1}{x^{2}} - 1} - \frac {3}{2} \, \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x \mathrm {sgn}\left (x\right ) + \frac {3}{2} \, \arcsin \left (x\right ) \mathrm {sgn}\left (x\right ) - \frac {x \mathrm {sgn}\left (x\right )}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\left (x\right )}{2 \, x} \]
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Time = 18.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )}{x} \, dx=\sqrt {\frac {1}{x^2}-1}-\frac {3\,\mathrm {atan}\left (\sqrt {\frac {1}{x^2}-1}\right )}{2}+\frac {x^2\,\sqrt {\frac {1}{x^2}-1}}{2} \]
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