Optimal. Leaf size=16 \[ \frac {\sqrt {1+x^2+x^4}}{x} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1604}
\begin {gather*} \frac {\sqrt {x^4+x^2+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 1604
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^2 \sqrt {1+x^2+x^4}} \, dx &=\frac {\sqrt {1+x^2+x^4}}{x}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^2+x^4}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 15, normalized size = 0.94
method | result | size |
default | \(\frac {\sqrt {x^{4}+x^{2}+1}}{x}\) | \(15\) |
trager | \(\frac {\sqrt {x^{4}+x^{2}+1}}{x}\) | \(15\) |
risch | \(\frac {\sqrt {x^{4}+x^{2}+1}}{x}\) | \(15\) |
elliptic | \(\frac {\sqrt {x^{4}+x^{2}+1}}{x}\) | \(15\) |
gosper | \(\frac {\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{\sqrt {x^{4}+x^{2}+1}\, x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.16, size = 22, normalized size = 1.38 \begin {gather*} \frac {\sqrt {x^{2} + x + 1} \sqrt {x^{2} - x + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 14, normalized size = 0.88 \begin {gather*} \frac {\sqrt {x^{4} + x^{2} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 14, normalized size = 0.88 \begin {gather*} \frac {\sqrt {x^4+x^2+1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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