Optimal. Leaf size=89 \[ -\frac {1}{2 x^2}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )-\frac {1}{4} \tan ^{-1}\left (2 x^2+\sqrt {3}\right )-\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{8 \sqrt {3}}+\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{8 \sqrt {3}} \]
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Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1490, 1281, 1127, 1161, 618, 204, 1164, 628} \begin {gather*} -\frac {1}{2 x^2}-\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{8 \sqrt {3}}+\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{8 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )-\frac {1}{4} \tan ^{-1}\left (2 x^2+\sqrt {3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 1127
Rule 1161
Rule 1164
Rule 1281
Rule 1490
Rubi steps
\begin {align*} \int \frac {1-x^4}{x^3 \left (1-x^4+x^8\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x^2}{x^2 \left (1-x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,x^2\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, 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 )}{8 \sqrt {3}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx,x,x^2\right )}{8 \sqrt {3}}\\ &=-\frac {1}{2 x^2}-\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x^2\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x^2\right )-\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.55 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\&,\frac {\text {$\#$1}^2 \log (x-\text {$\#$1})}{2 \text {$\#$1}^4-1}\&\right ]-\frac {1}{2 x^2} \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 {1-x^4}{x^3 \left (1-x^4+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.03, size = 188, normalized size = 2.11 \begin {gather*} \frac {4 \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2 \, x^{4} + \sqrt {6} \sqrt {2} x^{2} + 2} - \sqrt {3}\right ) + 4 \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2 \, x^{4} - \sqrt {6} \sqrt {2} x^{2} + 2} + \sqrt {3}\right ) + \sqrt {6} \sqrt {2} x^{2} \log \left (2 \, x^{4} + \sqrt {6} \sqrt {2} x^{2} + 2\right ) - \sqrt {6} \sqrt {2} x^{2} \log \left (2 \, x^{4} - \sqrt {6} \sqrt {2} x^{2} + 2\right ) - 24}{48 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 81, normalized size = 0.91 \begin {gather*} -\frac {1}{24} \, \sqrt {3} x^{4} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) + \frac {1}{24} \, \sqrt {3} x^{4} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) - \frac {1}{4} \, x^{4} \arctan \left (2 \, x^{2} + \sqrt {3}\right ) - \frac {1}{4} \, x^{4} \arctan \left (2 \, x^{2} - \sqrt {3}\right ) - \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.79 \begin {gather*} -\frac {\arctan \left (2 x^{2}-\sqrt {3}\right )}{4}-\frac {\arctan \left (2 x^{2}+\sqrt {3}\right )}{4}-\frac {\sqrt {3}\, \ln \left (x^{4}-\sqrt {3}\, x^{2}+1\right )}{24}+\frac {\sqrt {3}\, \ln \left (x^{4}+\sqrt {3}\, x^{2}+1\right )}{24}-\frac {1}{2 x^{2}} \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*} -\frac {1}{2 \, x^{2}} - \int \frac {x^{5}}{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 = 0.10, size = 56, normalized size = 0.63 \begin {gather*} \mathrm {atan}\left (\frac {2\,x^2}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\mathrm {atan}\left (\frac {2\,x^2}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {1}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 76, normalized size = 0.85 \begin {gather*} - \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{24} + \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{24} - \frac {\operatorname {atan}{\left (2 x^{2} - \sqrt {3} \right )}}{4} - \frac {\operatorname {atan}{\left (2 x^{2} + \sqrt {3} \right )}}{4} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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