Optimal. Leaf size=187 \[ -\frac {1}{8} \log \left (x^2-x+1\right ) (2 d-f+h)+\frac {1}{8} \log \left (x^2+x+1\right ) (2 d-f+h)+\frac {x \left (-\left (x^2 (d-2 f+h)\right )+d+f-2 h\right )}{6 \left (x^4+x^2+1\right )}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right ) (4 d+f+h)}{12 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) (4 d+f+h)}{12 \sqrt {3}}+\frac {(2 e-g) \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )} \]
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Rubi [A] time = 0.17, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {1673, 1678, 1169, 634, 618, 204, 628, 1247, 638} \begin {gather*} \frac {x \left (x^2 (-(d-2 f+h))+d+f-2 h\right )}{6 \left (x^4+x^2+1\right )}-\frac {1}{8} \log \left (x^2-x+1\right ) (2 d-f+h)+\frac {1}{8} \log \left (x^2+x+1\right ) (2 d-f+h)-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right ) (4 d+f+h)}{12 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) (4 d+f+h)}{12 \sqrt {3}}+\frac {x^2 (2 e-g)+e-2 g}{6 \left (x^4+x^2+1\right )}+\frac {(2 e-g) \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 638
Rule 1169
Rule 1247
Rule 1673
Rule 1678
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (1+x^2+x^4\right )^2} \, dx &=\int \frac {x \left (e+g x^2\right )}{\left (1+x^2+x^4\right )^2} \, dx+\int \frac {d+f x^2+h x^4}{\left (1+x^2+x^4\right )^2} \, dx\\ &=\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{6} \int \frac {5 d-f+2 h+(-d+2 f-h) x^2}{1+x^2+x^4} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e+g x}{\left (1+x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {e-2 g+(2 e-g) x^2}{6 \left (1+x^2+x^4\right )}+\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{12} \int \frac {5 d-f+2 h-(6 d-3 f+3 h) x}{1-x+x^2} \, dx+\frac {1}{12} \int \frac {5 d-f+2 h+(6 d-3 f+3 h) x}{1+x+x^2} \, dx+\frac {1}{6} (2 e-g) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {e-2 g+(2 e-g) x^2}{6 \left (1+x^2+x^4\right )}+\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{3} (-2 e+g) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )+\frac {1}{8} (-2 d+f-h) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{8} (2 d-f+h) \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {1}{24} (4 d+f+h) \int \frac {1}{1-x+x^2} \, dx+\frac {1}{24} (4 d+f+h) \int \frac {1}{1+x+x^2} \, dx\\ &=\frac {e-2 g+(2 e-g) x^2}{6 \left (1+x^2+x^4\right )}+\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {(2 e-g) \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{8} (2 d-f+h) \log \left (1-x+x^2\right )+\frac {1}{8} (2 d-f+h) \log \left (1+x+x^2\right )+\frac {1}{12} (-4 d-f-h) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{12} (-4 d-f-h) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {e-2 g+(2 e-g) x^2}{6 \left (1+x^2+x^4\right )}+\frac {x \left (d+f-2 h-(d-2 f+h) x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac {(4 d+f+h) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(4 d+f+h) \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(2 e-g) \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{8} (2 d-f+h) \log \left (1-x+x^2\right )+\frac {1}{8} (2 d-f+h) \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.61, size = 234, normalized size = 1.25 \begin {gather*} \frac {1}{36} \left (-\frac {6 \left (x \left (d \left (x^2-1\right )-f \left (2 x^2+1\right )+h \left (x^2+2\right )\right )-e \left (2 x^2+1\right )+g \left (x^2+2\right )\right )}{x^4+x^2+1}-\frac {\tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x\right ) \left (\left (\sqrt {3}-11 i\right ) d-2 \left (\sqrt {3}-2 i\right ) f+\left (\sqrt {3}-5 i\right ) h\right )}{\sqrt {\frac {1}{6} \left (1+i \sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x\right ) \left (\left (\sqrt {3}+11 i\right ) d-2 \left (\sqrt {3}+2 i\right ) f+\left (\sqrt {3}+5 i\right ) h\right )}{\sqrt {\frac {1}{6} \left (1-i \sqrt {3}\right )}}-4 \sqrt {3} (2 e-g) \tan ^{-1}\left (\frac {\sqrt {3}}{2 x^2+1}\right )\right ) \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+h x^4}{\left (1+x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 5.91, size = 255, normalized size = 1.36 \begin {gather*} -\frac {12 \, {\left (d - 2 \, f + h\right )} x^{3} - 12 \, {\left (2 \, e - g\right )} x^{2} - 2 \, \sqrt {3} {\left ({\left (4 \, d - 8 \, e + f + 4 \, g + h\right )} x^{4} + {\left (4 \, d - 8 \, e + f + 4 \, g + h\right )} x^{2} + 4 \, d - 8 \, e + f + 4 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (4 \, d + 8 \, e + f - 4 \, g + h\right )} x^{4} + {\left (4 \, d + 8 \, e + f - 4 \, g + h\right )} x^{2} + 4 \, d + 8 \, e + f - 4 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 12 \, {\left (d + f - 2 \, h\right )} x - 9 \, {\left ({\left (2 \, d - f + h\right )} x^{4} + {\left (2 \, d - f + h\right )} x^{2} + 2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) + 9 \, {\left ({\left (2 \, d - f + h\right )} x^{4} + {\left (2 \, d - f + h\right )} x^{2} + 2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - 12 \, e + 24 \, g}{72 \, {\left (x^{4} + x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 155, normalized size = 0.83 \begin {gather*} \frac {1}{36} \, \sqrt {3} {\left (4 \, d + f + 4 \, g + h - 8 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, d + f - 4 \, g + h + 8 \, e\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - \frac {d x^{3} - 2 \, f x^{3} + h x^{3} + g x^{2} - 2 \, x^{2} e - d x - f x + 2 \, h x + 2 \, g - e}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 328, normalized size = 1.75 \begin {gather*} \frac {\sqrt {3}\, d \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}+\frac {\sqrt {3}\, d \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}-\frac {d \ln \left (x^{2}-x +1\right )}{4}+\frac {d \ln \left (x^{2}+x +1\right )}{4}-\frac {2 \sqrt {3}\, e \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}+\frac {\sqrt {3}\, f \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{36}+\frac {\sqrt {3}\, f \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{36}+\frac {f \ln \left (x^{2}-x +1\right )}{8}-\frac {f \ln \left (x^{2}+x +1\right )}{8}+\frac {\sqrt {3}\, g \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}-\frac {\sqrt {3}\, g \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}+\frac {\sqrt {3}\, h \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{36}+\frac {\sqrt {3}\, h \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{36}-\frac {h \ln \left (x^{2}-x +1\right )}{8}+\frac {h \ln \left (x^{2}+x +1\right )}{8}+\frac {-\frac {2 d}{3}+\frac {e}{3}+\frac {f}{3}-\frac {2 g}{3}+\frac {h}{3}+\left (-\frac {d}{3}-\frac {e}{3}+\frac {2 f}{3}-\frac {g}{3}-\frac {h}{3}\right ) x}{4 x^{2}+4 x +4}-\frac {-\frac {2 d}{3}-\frac {e}{3}+\frac {f}{3}+\frac {2 g}{3}+\frac {h}{3}+\left (\frac {d}{3}-\frac {e}{3}-\frac {2 f}{3}-\frac {g}{3}+\frac {h}{3}\right ) x}{4 \left (x^{2}-x +1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 143, normalized size = 0.76 \begin {gather*} \frac {1}{36} \, \sqrt {3} {\left (4 \, d - 8 \, e + f + 4 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, d + 8 \, e + f - 4 \, g + h\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, d - f + h\right )} \log \left (x^{2} - x + 1\right ) - \frac {{\left (d - 2 \, f + h\right )} x^{3} - {\left (2 \, e - g\right )} x^{2} - {\left (d + f - 2 \, h\right )} x - e + 2 \, g}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 1547, normalized size = 8.27
result too large to display
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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