Optimal. Leaf size=9 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]
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Rubi [A] time = 0.0038201, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {25, 261} \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]
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 )^2} \, dx &=\int \frac{1}{\left (1+\frac{1}{x^2}\right )^{3/2} x^3} \, dx\\ &=\frac{1}{\sqrt{1+\frac{1}{x^2}}}\\ \end{align*}
Mathematica [B] time = 0.0062454, size = 20, normalized size = 2.22 \[ \frac{\sqrt{\frac{1}{x^2}+1} x^2}{x^2+1} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 23, normalized size = 2.6 \begin{align*}{\frac{{x}^{2}}{{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 = 1.54989, size = 15, normalized size = 1.67 \begin{align*} \frac{1}{\sqrt{\frac{x^{2} + 1}{x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44177, size = 63, normalized size = 7. \begin{align*} \frac{x^{2} \sqrt{\frac{x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.40479, size = 8, normalized size = 0.89 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12942, size = 15, normalized size = 1.67 \begin{align*} \frac{x \mathrm{sgn}\left (x\right )}{\sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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