Optimal. Leaf size=20 \[ \frac {\sqrt {x^2} \sqrt {x^2+1}}{x} \]
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Rubi [A] time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1958, 15, 261} \begin {gather*} \frac {\sqrt {x^2} \sqrt {x^2+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 261
Rule 1958
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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.85 \begin {gather*} \frac {x}{\sqrt {\frac {x^2}{x^2+1}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.61, size = 20, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^2} \sqrt {x^2+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 22, normalized size = 1.10 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \sqrt {\frac {x^{2}}{x^{2} + 1}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 15, normalized size = 0.75 \begin {gather*} \sqrt {x^{2} + 1} \mathrm {sgn}\relax (x) - \mathrm {sgn}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 1.15 \begin {gather*} \frac {\left (x^{2}+1\right ) \sqrt {\frac {x^{2}}{x^{2}+1}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 7, normalized size = 0.35 \begin {gather*} \sqrt {x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.41, size = 13, normalized size = 0.65 \begin {gather*} \frac {\sqrt {x^4+x^2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 36, normalized size = 1.80 \begin {gather*} x \sqrt {x^{2}} \sqrt {\frac {1}{x^{2} + 1}} + \frac {\sqrt {x^{2}} \sqrt {\frac {1}{x^{2} + 1}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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