Optimal. Leaf size=16 \[ \frac {\sqrt {x^4+x^2+1}}{x} \]
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Rubi [A] time = 0.02, 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 = {1590} \[ \frac {\sqrt {x^4+x^2+1}}{x} \]
Antiderivative was successfully verified.
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Rule 1590
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 = 0.03, size = 16, normalized size = 1.00 \[ \frac {\sqrt {x^4+x^2+1}}{x} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.80, size = 16, normalized size = 1.00 \[ \frac {\sqrt {x^4+x^2+1}}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 14, normalized size = 0.88 \[ \frac {\sqrt {x^{4} + x^{2} + 1}}{x} \]
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 \[ \int \frac {x^{4} - 1}{\sqrt {x^{4} + x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, 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 = 0.87, size = 22, normalized size = 1.38 \[ \frac {\sqrt {x^{2} + x + 1} \sqrt {x^{2} - x + 1}}{x} \]
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 \[ \frac {\sqrt {x^4+x^2+1}}{x} \]
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 \[ \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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