Optimal. Leaf size=12 \[ \frac {1}{2} \sinh ^{-1}\left (\frac {x^2}{2}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {275, 215} \[ \frac {1}{2} \sinh ^{-1}\left (\frac {x^2}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {4+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {4+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \sinh ^{-1}\left (\frac {x^2}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 12, normalized size = 1.00 \[ \frac {1}{2} \sinh ^{-1}\left (\frac {x^2}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 16, normalized size = 1.33 \[ -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 16, normalized size = 1.33 \[ -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 9, normalized size = 0.75 \[ \frac {\arcsinh \left (\frac {x^{2}}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 33, normalized size = 2.75 \[ \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 4}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 4}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 8, normalized size = 0.67 \[ \frac {\mathrm {asinh}\left (\frac {x^2}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 7, normalized size = 0.58 \[ \frac {\operatorname {asinh}{\left (\frac {x^{2}}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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