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

Optimal. Leaf size=104 \[ -\frac{1}{4} (d-f) \log \left (x^2-x+1\right )+\frac{1}{4} (d-f) \log \left (x^2+x+1\right )-\frac{(d+f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{(d+f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.18929, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{1}{4} (d-f) \log \left (x^2-x+1\right )+\frac{1}{4} (d-f) \log \left (x^2+x+1\right )-\frac{(d+f) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{(d+f) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.2623, size = 105, normalized size = 1.01 \[ \frac{\sqrt{3} e \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{3} - \left (\frac{d}{4} - \frac{f}{4}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{d}{4} - \frac{f}{4}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (d + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \left (d + f\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*e*atan(sqrt(3)*(2*x**2/3 + 1/3))/3 - (d/4 - f/4)*log(x**2 - x + 1) + (d/
4 - f/4)*log(x**2 + x + 1) + sqrt(3)*(d + f)*atan(sqrt(3)*(2*x/3 - 1/3))/6 + sqr
t(3)*(d + f)*atan(sqrt(3)*(2*x/3 + 1/3))/6

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Mathematica [C]  time = 0.270526, size = 121, normalized size = 1.16 \[ \frac{\left (2 i d+\left (\sqrt{3}-i\right ) f\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{6+6 i \sqrt{3}}}+\frac{\left (\left (\sqrt{3}+i\right ) f-2 i d\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{6-6 i \sqrt{3}}}-\frac{e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )}{\sqrt{3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(((2*I)*d + (-I + Sqrt[3])*f)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[6 + (6*I)*Sqrt[
3]] + (((-2*I)*d + (I + Sqrt[3])*f)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[6 - (6*I)*
Sqrt[3]] - (e*ArcTan[Sqrt[3]/(1 + 2*x^2)])/Sqrt[3]

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Maple [A]  time = 0.007, size = 148, normalized size = 1.4 \[{\frac{d\ln \left ({x}^{2}+x+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) f}{4}}+{\frac{d\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}e}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ({x}^{2}-x+1 \right ) f}{4}}-{\frac{d\ln \left ({x}^{2}-x+1 \right ) }{4}}+{\frac{d\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}e}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}f}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.790007, size = 101, normalized size = 0.97 \[ \frac{1}{6} \, \sqrt{3}{\left (d - 2 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (d + 2 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \,{\left (d - f\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{4} \,{\left (d - f\right )} \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.289862, size = 107, normalized size = 1.03 \[ \frac{1}{12} \, \sqrt{3}{\left (\sqrt{3}{\left (d - f\right )} \log \left (x^{2} + x + 1\right ) - \sqrt{3}{\left (d - f\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left (d - 2 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left (d + 2 \, e + f\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*(sqrt(3)*(d - f)*log(x^2 + x + 1) - sqrt(3)*(d - f)*log(x^2 - x + 1
) + 2*(d - 2*e + f)*arctan(1/3*sqrt(3)*(2*x + 1)) + 2*(d + 2*e + f)*arctan(1/3*s
qrt(3)*(2*x - 1)))

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Sympy [A]  time = 81.8656, size = 3589, normalized size = 34.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12)*log(x + (-7*d**5*e + 6*d**5*(-d/4 + f/
4 - sqrt(3)*I*(d + 2*e + f)/12) + 25*d**4*e*f + 18*d**4*f*(-d/4 + f/4 - sqrt(3)*
I*(d + 2*e + f)/12) - 15*d**3*e**3 - 18*d**3*e**2*(-d/4 + f/4 - sqrt(3)*I*(d + 2
*e + f)/12) - 25*d**3*e*f**2 + 60*d**3*e*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/1
2)**2 - 42*d**3*f**2*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12) + 72*d**3*(-d/4 +
 f/4 - sqrt(3)*I*(d + 2*e + f)/12)**3 + 108*d**2*e**2*f*(-d/4 + f/4 - sqrt(3)*I*
(d + 2*e + f)/12) + 20*d**2*e*f**3 - 144*d**2*e*f*(-d/4 + f/4 - sqrt(3)*I*(d + 2
*e + f)/12)**2 - 12*d**2*f**3*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12) - 144*d*
*2*f*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12)**3 + 4*d*e**5 + 24*d*e**4*(-d/4 +
 f/4 - sqrt(3)*I*(d + 2*e + f)/12) + 15*d*e**3*f**2 + 48*d*e**3*(-d/4 + f/4 - sq
rt(3)*I*(d + 2*e + f)/12)**2 - 54*d*e**2*f**2*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e +
 f)/12) + 288*d*e**2*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12)**3 - 20*d*e*f**4
+ 180*d*e*f**2*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12)**2 + 36*d*f**4*(-d/4 +
f/4 - sqrt(3)*I*(d + 2*e + f)/12) - 72*d*f**2*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e +
 f)/12)**3 - 8*e**5*f - 96*e**3*f*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12)**2 +
 36*e**2*f**3*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12) + 11*e*f**5 - 48*e*f**3*
(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12)**2 - 6*f**5*(-d/4 + f/4 - sqrt(3)*I*(d
 + 2*e + f)/12) + 144*f**3*(-d/4 + f/4 - sqrt(3)*I*(d + 2*e + f)/12)**3)/(3*d**6
 - 3*d**5*f - 8*d**4*e**2 - 3*d**4*f**2 + 40*d**3*e**2*f + 6*d**3*f**3 - 16*d**2
*e**4 - 48*d**2*e**2*f**2 - 3*d**2*f**4 + 16*d*e**4*f + 40*d*e**2*f**3 - 3*d*f**
5 - 16*e**4*f**2 - 8*e**2*f**4 + 3*f**6)) + (-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f
)/12)*log(x + (-7*d**5*e + 6*d**5*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) + 25
*d**4*e*f + 18*d**4*f*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) - 15*d**3*e**3 -
 18*d**3*e**2*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) - 25*d**3*e*f**2 + 60*d*
*3*e*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12)**2 - 42*d**3*f**2*(-d/4 + f/4 + s
qrt(3)*I*(d + 2*e + f)/12) + 72*d**3*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12)**
3 + 108*d**2*e**2*f*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) + 20*d**2*e*f**3 -
 144*d**2*e*f*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12)**2 - 12*d**2*f**3*(-d/4
+ f/4 + sqrt(3)*I*(d + 2*e + f)/12) - 144*d**2*f*(-d/4 + f/4 + sqrt(3)*I*(d + 2*
e + f)/12)**3 + 4*d*e**5 + 24*d*e**4*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) +
 15*d*e**3*f**2 + 48*d*e**3*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12)**2 - 54*d*
e**2*f**2*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) + 288*d*e**2*(-d/4 + f/4 + s
qrt(3)*I*(d + 2*e + f)/12)**3 - 20*d*e*f**4 + 180*d*e*f**2*(-d/4 + f/4 + sqrt(3)
*I*(d + 2*e + f)/12)**2 + 36*d*f**4*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) -
72*d*f**2*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12)**3 - 8*e**5*f - 96*e**3*f*(-
d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12)**2 + 36*e**2*f**3*(-d/4 + f/4 + sqrt(3)*
I*(d + 2*e + f)/12) + 11*e*f**5 - 48*e*f**3*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f
)/12)**2 - 6*f**5*(-d/4 + f/4 + sqrt(3)*I*(d + 2*e + f)/12) + 144*f**3*(-d/4 + f
/4 + sqrt(3)*I*(d + 2*e + f)/12)**3)/(3*d**6 - 3*d**5*f - 8*d**4*e**2 - 3*d**4*f
**2 + 40*d**3*e**2*f + 6*d**3*f**3 - 16*d**2*e**4 - 48*d**2*e**2*f**2 - 3*d**2*f
**4 + 16*d*e**4*f + 40*d*e**2*f**3 - 3*d*f**5 - 16*e**4*f**2 - 8*e**2*f**4 + 3*f
**6)) + (d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12)*log(x + (-7*d**5*e + 6*d**5*(d/
4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12) + 25*d**4*e*f + 18*d**4*f*(d/4 - f/4 - sqr
t(3)*I*(d - 2*e + f)/12) - 15*d**3*e**3 - 18*d**3*e**2*(d/4 - f/4 - sqrt(3)*I*(d
 - 2*e + f)/12) - 25*d**3*e*f**2 + 60*d**3*e*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f
)/12)**2 - 42*d**3*f**2*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12) + 72*d**3*(d/4
- f/4 - sqrt(3)*I*(d - 2*e + f)/12)**3 + 108*d**2*e**2*f*(d/4 - f/4 - sqrt(3)*I*
(d - 2*e + f)/12) + 20*d**2*e*f**3 - 144*d**2*e*f*(d/4 - f/4 - sqrt(3)*I*(d - 2*
e + f)/12)**2 - 12*d**2*f**3*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12) - 144*d**2
*f*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12)**3 + 4*d*e**5 + 24*d*e**4*(d/4 - f/4
 - sqrt(3)*I*(d - 2*e + f)/12) + 15*d*e**3*f**2 + 48*d*e**3*(d/4 - f/4 - sqrt(3)
*I*(d - 2*e + f)/12)**2 - 54*d*e**2*f**2*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12
) + 288*d*e**2*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12)**3 - 20*d*e*f**4 + 180*d
*e*f**2*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12)**2 + 36*d*f**4*(d/4 - f/4 - sqr
t(3)*I*(d - 2*e + f)/12) - 72*d*f**2*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12)**3
 - 8*e**5*f - 96*e**3*f*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12)**2 + 36*e**2*f*
*3*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12) + 11*e*f**5 - 48*e*f**3*(d/4 - f/4 -
 sqrt(3)*I*(d - 2*e + f)/12)**2 - 6*f**5*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12
) + 144*f**3*(d/4 - f/4 - sqrt(3)*I*(d - 2*e + f)/12)**3)/(3*d**6 - 3*d**5*f - 8
*d**4*e**2 - 3*d**4*f**2 + 40*d**3*e**2*f + 6*d**3*f**3 - 16*d**2*e**4 - 48*d**2
*e**2*f**2 - 3*d**2*f**4 + 16*d*e**4*f + 40*d*e**2*f**3 - 3*d*f**5 - 16*e**4*f**
2 - 8*e**2*f**4 + 3*f**6)) + (d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)*log(x + (-
7*d**5*e + 6*d**5*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12) + 25*d**4*e*f + 18*d*
*4*f*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12) - 15*d**3*e**3 - 18*d**3*e**2*(d/4
 - f/4 + sqrt(3)*I*(d - 2*e + f)/12) - 25*d**3*e*f**2 + 60*d**3*e*(d/4 - f/4 + s
qrt(3)*I*(d - 2*e + f)/12)**2 - 42*d**3*f**2*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f
)/12) + 72*d**3*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)**3 + 108*d**2*e**2*f*(d
/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12) + 20*d**2*e*f**3 - 144*d**2*e*f*(d/4 - f/
4 + sqrt(3)*I*(d - 2*e + f)/12)**2 - 12*d**2*f**3*(d/4 - f/4 + sqrt(3)*I*(d - 2*
e + f)/12) - 144*d**2*f*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)**3 + 4*d*e**5 +
 24*d*e**4*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12) + 15*d*e**3*f**2 + 48*d*e**3
*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)**2 - 54*d*e**2*f**2*(d/4 - f/4 + sqrt(
3)*I*(d - 2*e + f)/12) + 288*d*e**2*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)**3
- 20*d*e*f**4 + 180*d*e*f**2*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)**2 + 36*d*
f**4*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12) - 72*d*f**2*(d/4 - f/4 + sqrt(3)*I
*(d - 2*e + f)/12)**3 - 8*e**5*f - 96*e**3*f*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f
)/12)**2 + 36*e**2*f**3*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12) + 11*e*f**5 - 4
8*e*f**3*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)**2 - 6*f**5*(d/4 - f/4 + sqrt(
3)*I*(d - 2*e + f)/12) + 144*f**3*(d/4 - f/4 + sqrt(3)*I*(d - 2*e + f)/12)**3)/(
3*d**6 - 3*d**5*f - 8*d**4*e**2 - 3*d**4*f**2 + 40*d**3*e**2*f + 6*d**3*f**3 - 1
6*d**2*e**4 - 48*d**2*e**2*f**2 - 3*d**2*f**4 + 16*d*e**4*f + 40*d*e**2*f**3 - 3
*d*f**5 - 16*e**4*f**2 - 8*e**2*f**4 + 3*f**6))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273901, size = 104, normalized size = 1. \[ \frac{1}{6} \, \sqrt{3}{\left (d + f - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (d + f + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \,{\left (d - f\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{4} \,{\left (d - f\right )}{\rm ln}\left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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