Optimal. Leaf size=9 \[ \sqrt {\frac {1}{x^2}-1} \]
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Rubi [A] time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {25, 261} \[ \sqrt {\frac {1}{x^2}-1} \]
Antiderivative was successfully verified.
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Rule 25
Rule 261
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\frac {1}{x^2}}}{x \left (-1+x^2\right )} \, dx &=-\int \frac {1}{\sqrt {-1+\frac {1}{x^2}} x^3} \, dx\\ &=\sqrt {-1+\frac {1}{x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 9, normalized size = 1.00 \[ \sqrt {\frac {1}{x^2}-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 12, normalized size = 1.33 \[ \sqrt {-\frac {x^{2} - 1}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 37, normalized size = 4.11 \[ -\frac {x \mathrm {sgn}\relax (x)}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\relax (x)}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 1.44 \[ \sqrt {-\frac {x^{2}-1}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 16, normalized size = 1.78 \[ \frac {\sqrt {x + 1} \sqrt {-x + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.18, size = 14, normalized size = 1.56 \[ \frac {\sqrt {1-x^2}}{\relax |x|} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.25, size = 8, normalized size = 0.89 \[ \sqrt {-1 + \frac {1}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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