Optimal. Leaf size=11 \[ \frac {x}{1-x^2} \]
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Rubi [A] time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {28, 383} \[ \frac {x}{1-x^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 383
Rubi steps
\begin {align*} \int \frac {1+x^2}{1-2 x^2+x^4} \, dx &=\int \frac {1+x^2}{\left (-1+x^2\right )^2} \, dx\\ &=\frac {x}{1-x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 10, normalized size = 0.91 \[ -\frac {x}{x^2-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 10, normalized size = 0.91 \[ -\frac {x}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 11, normalized size = 1.00 \[ -\frac {1}{x - \frac {1}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 1.45 \[ -\frac {1}{2 \left (x +1\right )}-\frac {1}{2 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 10, normalized size = 0.91 \[ -\frac {x}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.34, size = 10, normalized size = 0.91 \[ -\frac {x}{x^2-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 7, normalized size = 0.64 \[ - \frac {x}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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