Optimal. Leaf size=127 \[ -\frac {1}{4} (d-f) \log \left (x^2-x+1\right )+\frac {1}{4} (d-f) \log \left (x^2+x+1\right )-\frac {(d+f) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(d+f) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(2 e-g) \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} g \log \left (x^4+x^2+1\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1673, 1169, 634, 618, 204, 628, 1247} \begin {gather*} -\frac {1}{4} (d-f) \log \left (x^2-x+1\right )+\frac {1}{4} (d-f) \log \left (x^2+x+1\right )-\frac {(d+f) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(d+f) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(2 e-g) \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} g \log \left (x^4+x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rule 1247
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx &=\int \frac {d+f x^2}{1+x^2+x^4} \, dx+\int \frac {x \left (e+g x^2\right )}{1+x^2+x^4} \, dx\\ &=\frac {1}{2} \int \frac {d-(d-f) x}{1-x+x^2} \, dx+\frac {1}{2} \int \frac {d+(d-f) x}{1+x+x^2} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e+g x}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} (d-f) \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {1}{4} (-d+f) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{4} (d+f) \int \frac {1}{1-x+x^2} \, dx+\frac {1}{4} (d+f) \int \frac {1}{1+x+x^2} \, dx+\frac {1}{4} (2 e-g) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )+\frac {1}{4} g \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{4} (d-f) \log \left (1-x+x^2\right )+\frac {1}{4} (d-f) \log \left (1+x+x^2\right )+\frac {1}{4} g \log \left (1+x^2+x^4\right )+\frac {1}{2} (-d-f) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{2} (-d-f) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+\frac {1}{2} (-2 e+g) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=-\frac {(d+f) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(d+f) \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(2 e-g) \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} (d-f) \log \left (1-x+x^2\right )+\frac {1}{4} (d-f) \log \left (1+x+x^2\right )+\frac {1}{4} g \log \left (1+x^2+x^4\right )\\ \end {align*}
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Mathematica [C] time = 0.48, size = 150, normalized size = 1.18 \begin {gather*} \frac {2 \left (\sqrt {2+2 i \sqrt {3}} \left (\left (\sqrt {3}+i\right ) f-2 i d\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x\right )+(2 g-4 e) \tan ^{-1}\left (\frac {\sqrt {3}}{2 x^2+1}\right )+\sqrt {3} g \log \left (x^4+x^2+1\right )\right )+2 \sqrt {2-2 i \sqrt {3}} \left (2 i d+\left (\sqrt {3}-i\right ) f\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x\right )}{8 \sqrt {3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.86, size = 83, normalized size = 0.65 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 85, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (d + f + g - 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + f - g + 2 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 204, normalized size = 1.61 \begin {gather*} \frac {\sqrt {3}\, d \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\sqrt {3}\, d \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {d \ln \left (x^{2}-x +1\right )}{4}+\frac {d \ln \left (x^{2}+x +1\right )}{4}-\frac {\sqrt {3}\, e \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, e \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, f \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\sqrt {3}\, f \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {f \ln \left (x^{2}-x +1\right )}{4}-\frac {f \ln \left (x^{2}+x +1\right )}{4}+\frac {\sqrt {3}\, g \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\sqrt {3}\, g \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {g \ln \left (x^{2}-x +1\right )}{4}+\frac {g \ln \left (x^{2}+x +1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 83, normalized size = 0.65 \begin {gather*} \frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 199, normalized size = 1.57 \begin {gather*} -\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}-\frac {f}{4}-\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right )-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{4}-\frac {d}{4}-\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{4}-\frac {d}{4}+\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}-\frac {f}{4}+\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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