Integrand size = 26, antiderivative size = 127 \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=-\frac {(d+f) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(d+f) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(2 e-g) \arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} (d-f) \log \left (1-x+x^2\right )+\frac {1}{4} (d-f) \log \left (1+x+x^2\right )+\frac {1}{4} g \log \left (1+x^2+x^4\right ) \]
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Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1687, 1183, 648, 632, 210, 642, 1261} \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right ) (d+f)}{2 \sqrt {3}}+\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right ) (d+f)}{2 \sqrt {3}}+\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right ) (2 e-g)}{2 \sqrt {3}}-\frac {1}{4} (d-f) \log \left (x^2-x+1\right )+\frac {1}{4} (d-f) \log \left (x^2+x+1\right )+\frac {1}{4} g \log \left (x^4+x^2+1\right ) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1261
Rule 1687
Rubi steps \begin{align*} \text {integral}& = \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} \text {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) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )+\frac {1}{4} g \text {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) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{2} (-d-f) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+\frac {1}{2} (-2 e+g) \text {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*}
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.18 \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=\frac {2 \sqrt {2-2 i \sqrt {3}} \left (2 i d+\left (-i+\sqrt {3}\right ) f\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )+2 \left (\sqrt {2+2 i \sqrt {3}} \left (-2 i d+\left (i+\sqrt {3}\right ) f\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )+(-4 e+2 g) \arctan \left (\frac {\sqrt {3}}{1+2 x^2}\right )+\sqrt {3} g \log \left (1+x^2+x^4\right )\right )}{8 \sqrt {3}} \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {\left (f -d +g \right ) \ln \left (x^{2}-x +1\right )}{4}+\frac {\left (\frac {d}{2}+e +\frac {f}{2}-\frac {g}{2}\right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\left (d -f +g \right ) \ln \left (x^{2}+x +1\right )}{4}+\frac {\left (\frac {d}{2}-e +\frac {f}{2}+\frac {g}{2}\right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(90\) |
risch | \(\text {Expression too large to display}\) | \(27628\) |
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Time = 0.41 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \]
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Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (d - 2 \, e + f + g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (d + 2 \, e + f - g\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (d - f + g\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (d - f - g\right )} \log \left (x^{2} - x + 1\right ) \]
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Time = 8.11 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.57 \[ \int \frac {d+e x+f x^2+g x^3}{1+x^2+x^4} \, dx=-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}-\frac {f}{4}-\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right )-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{4}-\frac {d}{4}-\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {f}{4}-\frac {d}{4}+\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {d}{4}-\frac {f}{4}+\frac {g}{4}+\frac {\sqrt {3}\,d\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,e\,1{}\mathrm {i}}{6}+\frac {\sqrt {3}\,f\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,g\,1{}\mathrm {i}}{12}\right ) \]
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