Optimal. Leaf size=20 \[ \frac {\sqrt {x^2} \sqrt {1+x^2}}{x} \]
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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 = {1978, 15, 267}
\begin {gather*} \frac {\sqrt {x^2} \sqrt {x^2+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 267
Rule 1978
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 = 0.00, size = 17, normalized size = 0.85 \begin {gather*} \frac {x}{\sqrt {\frac {x^2}{1+x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 23, normalized size = 1.15
method | result | size |
gosper | \(\frac {\left (x^{2}+1\right ) \sqrt {\frac {x^{2}}{x^{2}+1}}}{x}\) | \(23\) |
default | \(\frac {\left (x^{2}+1\right ) \sqrt {\frac {x^{2}}{x^{2}+1}}}{x}\) | \(23\) |
trager | \(\frac {\left (x^{2}+1\right ) \sqrt {\frac {x^{2}}{x^{2}+1}}}{x}\) | \(23\) |
risch | \(\frac {\left (x^{2}+1\right ) \sqrt {\frac {x^{2}}{x^{2}+1}}}{x}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, 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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Fricas [A]
time = 0.38, 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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Sympy [A]
time = 0.16, size = 26, normalized size = 1.30 \begin {gather*} x \sqrt {\frac {x^{2}}{x^{2} + 1}} + \frac {\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 = 1.92, size = 15, normalized size = 0.75 \begin {gather*} \sqrt {x^{2} + 1} \mathrm {sgn}\left (x\right ) - \mathrm {sgn}\left (x\right ) \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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