Optimal. Leaf size=12 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{x^2}{2}\right ) \]
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Rubi [A] time = 0.0037786, antiderivative size = 12, normalized size of antiderivative = 1., 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 275
Rule 215
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*}
Mathematica [A] time = 0.0022902, size = 12, normalized size = 1. \[ \frac{1}{2} \sinh ^{-1}\left (\frac{x^2}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{2}{\it Arcsinh} \left ({\frac{{x}^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.986686, size = 45, normalized size = 3.75 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46362, size = 43, normalized size = 3.58 \begin{align*} -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24951, size = 7, normalized size = 0.58 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{x^{2}}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18324, size = 22, normalized size = 1.83 \begin{align*} -\frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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