Optimal. Leaf size=20 \[ \frac{\sqrt{x^2} \sqrt{x^2+1}}{x} \]
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Rubi [A] time = 0.0044744, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1958, 15, 261} \[ \frac{\sqrt{x^2} \sqrt{x^2+1}}{x} \]
Antiderivative was successfully verified.
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Rule 1958
Rule 15
Rule 261
Rubi steps
\begin{align*} \int \sqrt{\frac{x^2}{1+x^2}} \, dx &=\int \frac{\sqrt{x^2}}{\sqrt{1+x^2}} \, dx\\ &=\frac{\sqrt{x^2} \int \frac{x}{\sqrt{1+x^2}} \, dx}{x}\\ &=\frac{\sqrt{x^2} \sqrt{1+x^2}}{x}\\ \end{align*}
Mathematica [A] time = 0.0053422, size = 17, normalized size = 0.85 \[ \frac{x}{\sqrt{\frac{x^2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}+1}{x}\sqrt{{\frac{{x}^{2}}{{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.87452, size = 9, normalized size = 0.45 \begin{align*} \sqrt{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61264, size = 45, normalized size = 2.25 \begin{align*} \frac{{\left (x^{2} + 1\right )} \sqrt{\frac{x^{2}}{x^{2} + 1}}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.440631, size = 36, normalized size = 1.8 \begin{align*} x \sqrt{x^{2}} \sqrt{\frac{1}{x^{2} + 1}} + \frac{\sqrt{x^{2}} \sqrt{\frac{1}{x^{2} + 1}}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23685, size = 20, normalized size = 1. \begin{align*} \sqrt{x^{2} + 1} \mathrm{sgn}\left (x\right ) - \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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