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3.32 \(\int \frac{d+e x+f x^2}{\left (1+x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=165 \[ -\frac{1}{8} (2 d-f) \log \left (x^2-x+1\right )+\frac{1}{8} (2 d-f) \log \left (x^2+x+1\right )+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{6 \left (x^4+x^2+1\right )}-\frac{(4 d+f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 d+f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )} \]

[Out]

(e*(1 + 2*x^2))/(6*(1 + x^2 + x^4)) + (x*(d + f - (d - 2*f)*x^2))/(6*(1 + x^2 +
x^4)) - ((4*d + f)*ArcTan[(1 - 2*x)/Sqrt[3]])/(12*Sqrt[3]) + ((4*d + f)*ArcTan[(
1 + 2*x)/Sqrt[3]])/(12*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^2)/Sqrt[3]])/(3*Sqrt[3])
- ((2*d - f)*Log[1 - x + x^2])/8 + ((2*d - f)*Log[1 + x + x^2])/8

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Rubi [A]  time = 0.296925, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476 \[ -\frac{1}{8} (2 d-f) \log \left (x^2-x+1\right )+\frac{1}{8} (2 d-f) \log \left (x^2+x+1\right )+\frac{x \left (x^2 (-(d-2 f))+d+f\right )}{6 \left (x^4+x^2+1\right )}-\frac{(4 d+f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{(4 d+f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2)/(1 + x^2 + x^4)^2,x]

[Out]

(e*(1 + 2*x^2))/(6*(1 + x^2 + x^4)) + (x*(d + f - (d - 2*f)*x^2))/(6*(1 + x^2 +
x^4)) - ((4*d + f)*ArcTan[(1 - 2*x)/Sqrt[3]])/(12*Sqrt[3]) + ((4*d + f)*ArcTan[(
1 + 2*x)/Sqrt[3]])/(12*Sqrt[3]) + (2*e*ArcTan[(1 + 2*x^2)/Sqrt[3]])/(3*Sqrt[3])
- ((2*d - f)*Log[1 - x + x^2])/8 + ((2*d - f)*Log[1 + x + x^2])/8

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Rubi in Sympy [A]  time = 52.5985, size = 144, normalized size = 0.87 \[ \frac{2 \sqrt{3} e \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{9} + \frac{x \left (d - e x^{3} + e x + f - x^{2} \left (d - 2 f\right )\right )}{6 \left (x^{4} + x^{2} + 1\right )} - \left (\frac{d}{4} - \frac{f}{8}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{d}{4} - \frac{f}{8}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (4 d + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{36} + \frac{\sqrt{3} \left (4 d + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**2+e*x+d)/(x**4+x**2+1)**2,x)

[Out]

2*sqrt(3)*e*atan(sqrt(3)*(2*x**2/3 + 1/3))/9 + x*(d - e*x**3 + e*x + f - x**2*(d
 - 2*f))/(6*(x**4 + x**2 + 1)) - (d/4 - f/8)*log(x**2 - x + 1) + (d/4 - f/8)*log
(x**2 + x + 1) + sqrt(3)*(4*d + f)*atan(sqrt(3)*(2*x/3 - 1/3))/36 + sqrt(3)*(4*d
 + f)*atan(sqrt(3)*(2*x/3 + 1/3))/36

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Mathematica [C]  time = 0.828077, size = 186, normalized size = 1.13 \[ \frac{1}{36} \left (\frac{6 \left (x \left (-d x^2+d+2 f x^2+f\right )+2 e x^2+e\right )}{x^4+x^2+1}-\frac{\left (\left (\sqrt{3}-11 i\right ) d-2 \left (\sqrt{3}-2 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1+i \sqrt{3}\right )}}-\frac{\left (\left (\sqrt{3}+11 i\right ) d-2 \left (\sqrt{3}+2 i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{\frac{1}{6} \left (1-i \sqrt{3}\right )}}-8 \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x + f*x^2)/(1 + x^2 + x^4)^2,x]

[Out]

((6*(e + 2*e*x^2 + x*(d + f - d*x^2 + 2*f*x^2)))/(1 + x^2 + x^4) - (((-11*I + Sq
rt[3])*d - 2*(-2*I + Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[(1 + I*Sqrt[
3])/6] - (((11*I + Sqrt[3])*d - 2*(2*I + Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2]
)/Sqrt[(1 - I*Sqrt[3])/6] - 8*Sqrt[3]*e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/36

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Maple [A]  time = 0.016, size = 214, normalized size = 1.3 \[{\frac{1}{4\,{x}^{2}+4\,x+4} \left ( \left ( -{\frac{d}{3}}-{\frac{e}{3}}+{\frac{2\,f}{3}} \right ) x-{\frac{2\,d}{3}}+{\frac{e}{3}}+{\frac{f}{3}} \right ) }+{\frac{d\ln \left ({x}^{2}+x+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{8}}+{\frac{d\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{36}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{2}-4\,x+4} \left ( \left ({\frac{d}{3}}-{\frac{e}{3}}-{\frac{2\,f}{3}} \right ) x-{\frac{2\,d}{3}}-{\frac{e}{3}}+{\frac{f}{3}} \right ) }-{\frac{d\ln \left ({x}^{2}-x+1 \right ) }{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{8}}+{\frac{d\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{36}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^2+e*x+d)/(x^4+x^2+1)^2,x)

[Out]

1/4*((-1/3*d-1/3*e+2/3*f)*x-2/3*d+1/3*e+1/3*f)/(x^2+x+1)+1/4*d*ln(x^2+x+1)-1/8*l
n(x^2+x+1)*f+1/9*d*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-2/9*3^(1/2)*arctan(1/3*(1
+2*x)*3^(1/2))*e+1/36*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*f-1/4*((1/3*d-1/3*e-2/
3*f)*x-2/3*d-1/3*e+1/3*f)/(x^2-x+1)-1/4*d*ln(x^2-x+1)+1/8*ln(x^2-x+1)*f+1/9*3^(1
/2)*arctan(1/3*(2*x-1)*3^(1/2))*d+2/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*e+1/36
*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*f

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Maxima [A]  time = 0.781035, size = 162, normalized size = 0.98 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, d - 8 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, d + 8 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, d - f\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, d - f\right )} \log \left (x^{2} - x + 1\right ) - \frac{{\left (d - 2 \, f\right )} x^{3} - 2 \, e x^{2} -{\left (d + f\right )} x - e}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 + x^2 + 1)^2,x, algorithm="maxima")

[Out]

1/36*sqrt(3)*(4*d - 8*e + f)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/36*sqrt(3)*(4*d +
 8*e + f)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*(2*d - f)*log(x^2 + x + 1) - 1/8*(
2*d - f)*log(x^2 - x + 1) - 1/6*((d - 2*f)*x^3 - 2*e*x^2 - (d + f)*x - e)/(x^4 +
 x^2 + 1)

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Fricas [A]  time = 0.311506, size = 297, normalized size = 1.8 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left ({\left (2 \, d - f\right )} x^{4} +{\left (2 \, d - f\right )} x^{2} + 2 \, d - f\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left ({\left (2 \, d - f\right )} x^{4} +{\left (2 \, d - f\right )} x^{2} + 2 \, d - f\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left ({\left (4 \, d - 8 \, e + f\right )} x^{4} +{\left (4 \, d - 8 \, e + f\right )} x^{2} + 4 \, d - 8 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left ({\left (4 \, d + 8 \, e + f\right )} x^{4} +{\left (4 \, d + 8 \, e + f\right )} x^{2} + 4 \, d + 8 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 4 \, \sqrt{3}{\left ({\left (d - 2 \, f\right )} x^{3} - 2 \, e x^{2} -{\left (d + f\right )} x - e\right )}\right )}}{72 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 + x^2 + 1)^2,x, algorithm="fricas")

[Out]

1/72*sqrt(3)*(3*sqrt(3)*((2*d - f)*x^4 + (2*d - f)*x^2 + 2*d - f)*log(x^2 + x +
1) - 3*sqrt(3)*((2*d - f)*x^4 + (2*d - f)*x^2 + 2*d - f)*log(x^2 - x + 1) + 2*((
4*d - 8*e + f)*x^4 + (4*d - 8*e + f)*x^2 + 4*d - 8*e + f)*arctan(1/3*sqrt(3)*(2*
x + 1)) + 2*((4*d + 8*e + f)*x^4 + (4*d + 8*e + f)*x^2 + 4*d + 8*e + f)*arctan(1
/3*sqrt(3)*(2*x - 1)) - 4*sqrt(3)*((d - 2*f)*x^3 - 2*e*x^2 - (d + f)*x - e))/(x^
4 + x^2 + 1)

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Sympy [A]  time = 91.8248, size = 4107, normalized size = 24.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**2+e*x+d)/(x**4+x**2+1)**2,x)

[Out]

(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)*log(x + (-164944*d**5*e + 16416*d**5
*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 336520*d**4*e*f + 200664*d**4*f*(
-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) - 115200*d**3*e**3 - 504576*d**3*e**2
*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) - 272380*d**3*e*f**2 + 1734912*d**3
*e*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 - 229500*d**3*f**2*(-d/4 + f/8
 - sqrt(3)*I*(4*d + 8*e + f)/72) + 2612736*d**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8
*e + f)/72)**3 + 51840*d**2*e**3*f + 881280*d**2*e**2*f*(-d/4 + f/8 - sqrt(3)*I*
(4*d + 8*e + f)/72) + 119420*d**2*e*f**3 - 2477952*d**2*e*f*(-d/4 + f/8 - sqrt(3
)*I*(4*d + 8*e + f)/72)**2 + 50436*d**2*f**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e
+ f)/72) - 2519424*d**2*f*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 28672
*d*e**5 + 184320*d*e**4*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 8640*d*e**
3*f**2 + 774144*d*e**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 - 409536*d
*e**2*f**2*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) + 4976640*d*e**2*(-d/4 +
f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3 - 31040*d*e*f**4 + 1270080*d*e*f**2*(-d/4
 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 + 14040*d*f**4*(-d/4 + f/8 - sqrt(3)*I
*(4*d + 8*e + f)/72) + 139968*d*f**2*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)
**3 - 20480*e**5*f - 36864*e**4*f*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) -
2880*e**3*f**3 - 552960*e**3*f*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 +
70848*e**2*f**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72) - 995328*e**2*f*(-d/
4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 3956*e*f**5 - 209088*e*f**3*(-d/4 +
 f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**2 - 3996*f**5*(-d/4 + f/8 - sqrt(3)*I*(4*d
 + 8*e + f)/72) + 233280*f**3*(-d/4 + f/8 - sqrt(3)*I*(4*d + 8*e + f)/72)**3)/(5
3568*d**6 - 69984*d**5*f - 182528*d**4*e**2 + 23652*d**4*f**2 + 377344*d**3*e**2
*f + 5400*d**3*f**3 - 126976*d**2*e**4 - 278400*d**2*e**2*f**2 - 4131*d**2*f**4
+ 102400*d*e**4*f + 93568*d*e**2*f**3 + 81*d*f**5 - 28672*e**4*f**2 - 11648*e**2
*f**4 + 189*f**6)) + (-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)*log(x + (-16494
4*d**5*e + 16416*d**5*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 336520*d**4*
e*f + 200664*d**4*f*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) - 115200*d**3*e*
*3 - 504576*d**3*e**2*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) - 272380*d**3*
e*f**2 + 1734912*d**3*e*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**2 - 229500*
d**3*f**2*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 2612736*d**3*(-d/4 + f/8
 + sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 51840*d**2*e**3*f + 881280*d**2*e**2*f*(-d
/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 119420*d**2*e*f**3 - 2477952*d**2*e*f
*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**2 + 50436*d**2*f**3*(-d/4 + f/8 +
sqrt(3)*I*(4*d + 8*e + f)/72) - 2519424*d**2*f*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*
e + f)/72)**3 + 28672*d*e**5 + 184320*d*e**4*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e
+ f)/72) + 8640*d*e**3*f**2 + 774144*d*e**3*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e +
 f)/72)**2 - 409536*d*e**2*f**2*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 49
76640*d*e**2*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**3 - 31040*d*e*f**4 + 1
270080*d*e*f**2*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**2 + 14040*d*f**4*(-
d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 139968*d*f**2*(-d/4 + f/8 + sqrt(3)*
I*(4*d + 8*e + f)/72)**3 - 20480*e**5*f - 36864*e**4*f*(-d/4 + f/8 + sqrt(3)*I*(
4*d + 8*e + f)/72) - 2880*e**3*f**3 - 552960*e**3*f*(-d/4 + f/8 + sqrt(3)*I*(4*d
 + 8*e + f)/72)**2 + 70848*e**2*f**3*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)
 - 995328*e**2*f*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**3 + 3956*e*f**5 -
209088*e*f**3*(-d/4 + f/8 + sqrt(3)*I*(4*d + 8*e + f)/72)**2 - 3996*f**5*(-d/4 +
 f/8 + sqrt(3)*I*(4*d + 8*e + f)/72) + 233280*f**3*(-d/4 + f/8 + sqrt(3)*I*(4*d
+ 8*e + f)/72)**3)/(53568*d**6 - 69984*d**5*f - 182528*d**4*e**2 + 23652*d**4*f*
*2 + 377344*d**3*e**2*f + 5400*d**3*f**3 - 126976*d**2*e**4 - 278400*d**2*e**2*f
**2 - 4131*d**2*f**4 + 102400*d*e**4*f + 93568*d*e**2*f**3 + 81*d*f**5 - 28672*e
**4*f**2 - 11648*e**2*f**4 + 189*f**6)) + (d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)
/72)*log(x + (-164944*d**5*e + 16416*d**5*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)
/72) + 336520*d**4*e*f + 200664*d**4*f*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72
) - 115200*d**3*e**3 - 504576*d**3*e**2*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/7
2) - 272380*d**3*e*f**2 + 1734912*d**3*e*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/
72)**2 - 229500*d**3*f**2*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72) + 2612736*d
**3*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**3 + 51840*d**2*e**3*f + 881280*d
**2*e**2*f*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72) + 119420*d**2*e*f**3 - 247
7952*d**2*e*f*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**2 + 50436*d**2*f**3*(d
/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72) - 2519424*d**2*f*(d/4 - f/8 - sqrt(3)*I
*(4*d - 8*e + f)/72)**3 + 28672*d*e**5 + 184320*d*e**4*(d/4 - f/8 - sqrt(3)*I*(4
*d - 8*e + f)/72) + 8640*d*e**3*f**2 + 774144*d*e**3*(d/4 - f/8 - sqrt(3)*I*(4*d
 - 8*e + f)/72)**2 - 409536*d*e**2*f**2*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/7
2) + 4976640*d*e**2*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**3 - 31040*d*e*f*
*4 + 1270080*d*e*f**2*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**2 + 14040*d*f*
*4*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72) + 139968*d*f**2*(d/4 - f/8 - sqrt(
3)*I*(4*d - 8*e + f)/72)**3 - 20480*e**5*f - 36864*e**4*f*(d/4 - f/8 - sqrt(3)*I
*(4*d - 8*e + f)/72) - 2880*e**3*f**3 - 552960*e**3*f*(d/4 - f/8 - sqrt(3)*I*(4*
d - 8*e + f)/72)**2 + 70848*e**2*f**3*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)
 - 995328*e**2*f*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**3 + 3956*e*f**5 - 2
09088*e*f**3*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*e + f)/72)**2 - 3996*f**5*(d/4 - f/
8 - sqrt(3)*I*(4*d - 8*e + f)/72) + 233280*f**3*(d/4 - f/8 - sqrt(3)*I*(4*d - 8*
e + f)/72)**3)/(53568*d**6 - 69984*d**5*f - 182528*d**4*e**2 + 23652*d**4*f**2 +
 377344*d**3*e**2*f + 5400*d**3*f**3 - 126976*d**2*e**4 - 278400*d**2*e**2*f**2
- 4131*d**2*f**4 + 102400*d*e**4*f + 93568*d*e**2*f**3 + 81*d*f**5 - 28672*e**4*
f**2 - 11648*e**2*f**4 + 189*f**6)) + (d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)
*log(x + (-164944*d**5*e + 16416*d**5*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)
 + 336520*d**4*e*f + 200664*d**4*f*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) -
115200*d**3*e**3 - 504576*d**3*e**2*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) -
 272380*d**3*e*f**2 + 1734912*d**3*e*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)*
*2 - 229500*d**3*f**2*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) + 2612736*d**3*
(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**3 + 51840*d**2*e**3*f + 881280*d**2*
e**2*f*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) + 119420*d**2*e*f**3 - 2477952
*d**2*e*f*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**2 + 50436*d**2*f**3*(d/4 -
 f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) - 2519424*d**2*f*(d/4 - f/8 + sqrt(3)*I*(4*
d - 8*e + f)/72)**3 + 28672*d*e**5 + 184320*d*e**4*(d/4 - f/8 + sqrt(3)*I*(4*d -
 8*e + f)/72) + 8640*d*e**3*f**2 + 774144*d*e**3*(d/4 - f/8 + sqrt(3)*I*(4*d - 8
*e + f)/72)**2 - 409536*d*e**2*f**2*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) +
 4976640*d*e**2*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**3 - 31040*d*e*f**4 +
 1270080*d*e*f**2*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**2 + 14040*d*f**4*(
d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) + 139968*d*f**2*(d/4 - f/8 + sqrt(3)*I
*(4*d - 8*e + f)/72)**3 - 20480*e**5*f - 36864*e**4*f*(d/4 - f/8 + sqrt(3)*I*(4*
d - 8*e + f)/72) - 2880*e**3*f**3 - 552960*e**3*f*(d/4 - f/8 + sqrt(3)*I*(4*d -
8*e + f)/72)**2 + 70848*e**2*f**3*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72) - 9
95328*e**2*f*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**3 + 3956*e*f**5 - 20908
8*e*f**3*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e + f)/72)**2 - 3996*f**5*(d/4 - f/8 +
sqrt(3)*I*(4*d - 8*e + f)/72) + 233280*f**3*(d/4 - f/8 + sqrt(3)*I*(4*d - 8*e +
f)/72)**3)/(53568*d**6 - 69984*d**5*f - 182528*d**4*e**2 + 23652*d**4*f**2 + 377
344*d**3*e**2*f + 5400*d**3*f**3 - 126976*d**2*e**4 - 278400*d**2*e**2*f**2 - 41
31*d**2*f**4 + 102400*d*e**4*f + 93568*d*e**2*f**3 + 81*d*f**5 - 28672*e**4*f**2
 - 11648*e**2*f**4 + 189*f**6)) - (-2*e*x**2 - e + x**3*(d - 2*f) + x*(-d - f))/
(6*x**4 + 6*x**2 + 6)

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GIAC/XCAS [A]  time = 0.280608, size = 173, normalized size = 1.05 \[ \frac{1}{36} \, \sqrt{3}{\left (4 \, d + f - 8 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{36} \, \sqrt{3}{\left (4 \, d + f + 8 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \,{\left (2 \, d - f\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{8} \,{\left (2 \, d - f\right )}{\rm ln}\left (x^{2} - x + 1\right ) - \frac{d x^{3} - 2 \, f x^{3} - 2 \, x^{2} e - d x - f x - e}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e*x + d)/(x^4 + x^2 + 1)^2,x, algorithm="giac")

[Out]

1/36*sqrt(3)*(4*d + f - 8*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/36*sqrt(3)*(4*d +
 f + 8*e)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*(2*d - f)*ln(x^2 + x + 1) - 1/8*(2
*d - f)*ln(x^2 - x + 1) - 1/6*(d*x^3 - 2*f*x^3 - 2*x^2*e - d*x - f*x - e)/(x^4 +
 x^2 + 1)

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