Optimal. Leaf size=21 \[ \frac{1}{\sqrt{\frac{1}{x^2}-1}}-\sqrt{\frac{1}{x^2}-1} \]
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Rubi [A] time = 0.0107285, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {25, 266, 43} \[ \frac{1}{\sqrt{\frac{1}{x^2}-1}}-\sqrt{\frac{1}{x^2}-1} \]
Antiderivative was successfully verified.
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Rule 25
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+\frac{1}{x^2}}}{x \left (-1+x^2\right )^2} \, dx &=\int \frac{1}{\left (-1+\frac{1}{x^2}\right )^{3/2} x^5} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(-1+x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^{3/2}}+\frac{1}{\sqrt{-1+x}}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{\sqrt{-1+\frac{1}{x^2}}}-\sqrt{-1+\frac{1}{x^2}}\\ \end{align*}
Mathematica [A] time = 0.0057347, size = 24, normalized size = 1.14 \[ \frac{\sqrt{\frac{1}{x^2}-1} \left (1-2 x^2\right )}{x^2-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 1.4 \begin{align*} -{\frac{2\,{x}^{2}-1}{{x}^{2}-1}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04155, size = 41, normalized size = 1.95 \begin{align*} -\frac{{\left (2 \, x^{2} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1}}{x^{3} - x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45863, size = 61, normalized size = 2.9 \begin{align*} -\frac{{\left (2 \, x^{2} - 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}}}{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.16238, size = 20, normalized size = 0.95 \begin{align*} - \sqrt{-1 + \frac{1}{x^{2}}} + \frac{1}{\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 = 1.17584, size = 78, normalized size = 3.71 \begin{align*} -\frac{\sqrt{-x^{2} + 1} x \mathrm{sgn}\left (x\right )}{x^{2} - 1} + \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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