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 = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1164
Rule 1490
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*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.88 \begin {gather*} \frac {\log \left (x^4+\sqrt {3} x^2+1\right )-\log \left (-x^4+\sqrt {3} x^2-1\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (1-x^4\right )}{1-x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.87, size = 41, normalized size = 0.82 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 31, normalized size = 0.62 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 39, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {3}\, \ln \left (x^{4}-\sqrt {3}\, x^{2}+1\right )}{12}+\frac {\sqrt {3}\, \ln \left (x^{4}+\sqrt {3}\, x^{2}+1\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{4} - 1\right )} x}{x^{8} - x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.89, size = 20, normalized size = 0.40 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x^2}{x^4+1}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 42, normalized size = 0.84 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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