Optimal. Leaf size=9 \[ \sqrt{\frac{1}{x^2}-1} \]
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Rubi [A] time = 0.0040838, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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*}
Mathematica [A] time = 0.0025487, size = 9, normalized size = 1. \[ \sqrt{\frac{1}{x^2}-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 13, normalized size = 1.4 \begin{align*} \sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.50257, size = 22, normalized size = 2.44 \begin{align*} \frac{\sqrt{x + 1} \sqrt{-x + 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4282, size = 30, normalized size = 3.33 \begin{align*} \sqrt{-\frac{x^{2} - 1}{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.88333, size = 8, normalized size = 0.89 \begin{align*} \sqrt{-1 + \frac{1}{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16072, size = 50, normalized size = 5.56 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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