Optimal. Leaf size=50 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]
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Rubi [A] time = 0.0399463, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1490, 1164, 628} \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1490
Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (1-x^4\right )}{1-x^4+x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{-1-\sqrt{3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}-2 x}{-1+\sqrt{3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt{3}}\\ &=-\frac{\log \left (1-\sqrt{3} x^2+x^4\right )}{4 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} x^2+x^4\right )}{4 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0161647, size = 44, normalized size = 0.88 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )-\log \left (-x^4+\sqrt{3} x^2-1\right )}{4 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 39, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (x^{4} - 1\right )} x}{x^{8} - x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72637, size = 104, normalized size = 2.08 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{x^{8} + 5 \, x^{4} + 2 \, \sqrt{3}{\left (x^{6} + x^{2}\right )} + 1}{x^{8} - x^{4} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.162072, size = 42, normalized size = 0.84 \begin{align*} - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11162, size = 42, normalized size = 0.84 \begin{align*} -\frac{1}{12} \, \sqrt{3} \log \left (\frac{x^{2} - \sqrt{3} + \frac{1}{x^{2}}}{x^{2} + \sqrt{3} + \frac{1}{x^{2}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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