Optimal. Leaf size=119 \[ -\frac {1}{8} (2 a-b) \log \left (x^2-x+1\right )+\frac {1}{8} (2 a-b) \log \left (x^2+x+1\right )+\frac {x \left (-\left (x^2 (a-2 b)\right )+a+b\right )}{6 \left (x^4+x^2+1\right )}-\frac {(4 a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(4 a+b) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{12 \sqrt {3}} \]
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Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1178, 1169, 634, 618, 204, 628} \begin {gather*} \frac {x \left (x^2 (-(a-2 b))+a+b\right )}{6 \left (x^4+x^2+1\right )}-\frac {1}{8} (2 a-b) \log \left (x^2-x+1\right )+\frac {1}{8} (2 a-b) \log \left (x^2+x+1\right )-\frac {(4 a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(4 a+b) \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{12 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rule 1178
Rubi steps
\begin {align*} \int \frac {a+b x^2}{\left (1+x^2+x^4\right )^2} \, dx &=\frac {x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{6} \int \frac {5 a-b+(-a+2 b) x^2}{1+x^2+x^4} \, dx\\ &=\frac {x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{12} \int \frac {5 a-b-(6 a-3 b) x}{1-x+x^2} \, dx+\frac {1}{12} \int \frac {5 a-b+(6 a-3 b) x}{1+x+x^2} \, dx\\ &=\frac {x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}+\frac {1}{8} (2 a-b) \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {1}{8} (-2 a+b) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{24} (4 a+b) \int \frac {1}{1-x+x^2} \, dx+\frac {1}{24} (4 a+b) \int \frac {1}{1+x+x^2} \, dx\\ &=\frac {x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac {1}{8} (2 a-b) \log \left (1-x+x^2\right )+\frac {1}{8} (2 a-b) \log \left (1+x+x^2\right )+\frac {1}{12} (-4 a-b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{12} (-4 a-b) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {x \left (a+b-(a-2 b) x^2\right )}{6 \left (1+x^2+x^4\right )}-\frac {(4 a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}+\frac {(4 a+b) \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{12 \sqrt {3}}-\frac {1}{8} (2 a-b) \log \left (1-x+x^2\right )+\frac {1}{8} (2 a-b) \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.25, size = 147, normalized size = 1.24 \begin {gather*} \frac {x \left (-a x^2+a+2 b x^2+b\right )}{6 \left (x^4+x^2+1\right )}-\frac {\left (\left (\sqrt {3}-11 i\right ) a-2 \left (\sqrt {3}-2 i\right ) b\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x\right )}{6 \sqrt {6+6 i \sqrt {3}}}-\frac {\left (\left (\sqrt {3}+11 i\right ) a-2 \left (\sqrt {3}+2 i\right ) b\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x\right )}{6 \sqrt {6-6 i \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 {a+b x^2}{\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 = 0.80, size = 185, normalized size = 1.55 \begin {gather*} -\frac {12 \, {\left (a - 2 \, b\right )} x^{3} - 2 \, \sqrt {3} {\left ({\left (4 \, a + b\right )} x^{4} + {\left (4 \, a + b\right )} x^{2} + 4 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, \sqrt {3} {\left ({\left (4 \, a + b\right )} x^{4} + {\left (4 \, a + b\right )} x^{2} + 4 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 12 \, {\left (a + b\right )} x - 9 \, {\left ({\left (2 \, a - b\right )} x^{4} + {\left (2 \, a - b\right )} x^{2} + 2 \, a - b\right )} \log \left (x^{2} + x + 1\right ) + 9 \, {\left ({\left (2 \, a - b\right )} x^{4} + {\left (2 \, a - b\right )} x^{2} + 2 \, a - b\right )} \log \left (x^{2} - x + 1\right )}{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 = 0.16, size = 109, normalized size = 0.92 \begin {gather*} \frac {1}{36} \, \sqrt {3} {\left (4 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) - \frac {a x^{3} - 2 \, b x^{3} - a x - b x}{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 = 0.01, size = 168, normalized size = 1.41 \begin {gather*} \frac {\sqrt {3}\, a \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}+\frac {\sqrt {3}\, a \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}-\frac {a \ln \left (x^{2}-x +1\right )}{4}+\frac {a \ln \left (x^{2}+x +1\right )}{4}+\frac {\sqrt {3}\, b \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{36}+\frac {\sqrt {3}\, b \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{36}+\frac {b \ln \left (x^{2}-x +1\right )}{8}-\frac {b \ln \left (x^{2}+x +1\right )}{8}+\frac {-\frac {2 a}{3}+\frac {b}{3}+\left (-\frac {a}{3}+\frac {2 b}{3}\right ) x}{4 x^{2}+4 x +4}-\frac {-\frac {2 a}{3}+\frac {b}{3}+\left (\frac {a}{3}-\frac {2 b}{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 = 2.35, size = 105, normalized size = 0.88 \begin {gather*} \frac {1}{36} \, \sqrt {3} {\left (4 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{36} \, \sqrt {3} {\left (4 \, a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, {\left (2 \, a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, {\left (2 \, a - b\right )} \log \left (x^{2} - x + 1\right ) - \frac {{\left (a - 2 \, b\right )} x^{3} - {\left (a + b\right )} x}{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 = 4.49, size = 897, normalized size = 7.54
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.89, size = 874, normalized size = 7.34
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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