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 = {281, 221}
\begin {gather*} \frac {1}{2} \sinh ^{-1}\left (\frac {x^2}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {4+x^4}} \, dx &=\frac {1}{2} \text {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 = 0.08, size = 18, normalized size = 1.50 \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {4+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 9, normalized size = 0.75
method | result | size |
default | \(\frac {\arcsinh \left (\frac {x^{2}}{2}\right )}{2}\) | \(9\) |
meijerg | \(\frac {\arcsinh \left (\frac {x^{2}}{2}\right )}{2}\) | \(9\) |
elliptic | \(\frac {\arcsinh \left (\frac {x^{2}}{2}\right )}{2}\) | \(9\) |
trager | \(-\frac {\ln \left (x^{2}-\sqrt {x^{4}+4}\right )}{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (8) = 16\).
time = 0.28, size = 33, normalized size = 2.75 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 16, normalized size = 1.33 \begin {gather*} -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 7, normalized size = 0.58 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {x^{2}}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 16, normalized size = 1.33 \begin {gather*} -\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 4}\right ) \end {gather*}
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 \begin {gather*} \frac {\mathrm {asinh}\left (\frac {x^2}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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