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

Optimal. Leaf size=140 \[ -\frac{1}{4} d \log \left (x^2-x+1\right )+\frac{1}{4} d \log \left (x^2+x+1\right )+\frac{d x \left (1-x^2\right )}{6 \left (x^4+x^2+1\right )}-\frac{d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )} \]

[Out]

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

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Rubi [A]  time = 0.218017, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625 \[ -\frac{1}{4} d \log \left (x^2-x+1\right )+\frac{1}{4} d \log \left (x^2+x+1\right )+\frac{d x \left (1-x^2\right )}{6 \left (x^4+x^2+1\right )}-\frac{d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{2 e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{e \left (2 x^2+1\right )}{6 \left (x^4+x^2+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.6818, size = 126, normalized size = 0.9 \[ - \frac{d \log{\left (x^{2} - x + 1 \right )}}{4} + \frac{d \log{\left (x^{2} + x + 1 \right )}}{4} + \frac{\sqrt{3} d \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{9} + \frac{\sqrt{3} d \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{9} + \frac{2 \sqrt{3} e \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{9} + \frac{x \left (- d x^{2} + d - e x^{3} + e x\right )}{6 \left (x^{4} + x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.986055, size = 146, normalized size = 1.04 \[ \frac{d \left (x-x^3\right )+2 e x^2+e}{6 \left (x^4+x^2+1\right )}-\frac{\left (\sqrt{3}-11 i\right ) d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{6 \sqrt{6+6 i \sqrt{3}}}-\frac{\left (\sqrt{3}+11 i\right ) d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{6 \sqrt{6-6 i \sqrt{3}}}-\frac{2 e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )}{3 \sqrt{3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.016, size = 146, normalized size = 1. \[{\frac{1}{4\,{x}^{2}+4\,x+4} \left ( \left ( -{\frac{d}{3}}-{\frac{e}{3}} \right ) x-{\frac{2\,d}{3}}+{\frac{e}{3}} \right ) }+{\frac{d\ln \left ({x}^{2}+x+1 \right ) }{4}}+{\frac{d\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{2}-4\,x+4} \left ( \left ({\frac{d}{3}}-{\frac{e}{3}} \right ) x-{\frac{2\,d}{3}}-{\frac{e}{3}} \right ) }-{\frac{d\ln \left ({x}^{2}-x+1 \right ) }{4}}+{\frac{d\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\sqrt{3}e}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(x^4+x^2+1)^2,x)

[Out]

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

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Maxima [A]  time = 0.777721, size = 130, normalized size = 0.93 \[ \frac{1}{9} \, \sqrt{3}{\left (d - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{9} \, \sqrt{3}{\left (d + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \, d \log \left (x^{2} + x + 1\right ) - \frac{1}{4} \, d \log \left (x^{2} - x + 1\right ) - \frac{d x^{3} - 2 \, e x^{2} - d x - e}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(3)*(d - 2*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/9*sqrt(3)*(d + 2*e)*arct
an(1/3*sqrt(3)*(2*x - 1)) + 1/4*d*log(x^2 + x + 1) - 1/4*d*log(x^2 - x + 1) - 1/
6*(d*x^3 - 2*e*x^2 - d*x - e)/(x^4 + x^2 + 1)

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Fricas [A]  time = 0.273321, size = 219, normalized size = 1.56 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (d x^{4} + d x^{2} + d\right )} \log \left (x^{2} + x + 1\right ) - 3 \, \sqrt{3}{\left (d x^{4} + d x^{2} + d\right )} \log \left (x^{2} - x + 1\right ) + 4 \,{\left ({\left (d - 2 \, e\right )} x^{4} +{\left (d - 2 \, e\right )} x^{2} + d - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 4 \,{\left ({\left (d + 2 \, e\right )} x^{4} +{\left (d + 2 \, e\right )} x^{2} + d + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 2 \, \sqrt{3}{\left (d x^{3} - 2 \, e x^{2} - d x - e\right )}\right )}}{36 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*sqrt(3)*(3*sqrt(3)*(d*x^4 + d*x^2 + d)*log(x^2 + x + 1) - 3*sqrt(3)*(d*x^4
+ d*x^2 + d)*log(x^2 - x + 1) + 4*((d - 2*e)*x^4 + (d - 2*e)*x^2 + d - 2*e)*arct
an(1/3*sqrt(3)*(2*x + 1)) + 4*((d + 2*e)*x^4 + (d + 2*e)*x^2 + d + 2*e)*arctan(1
/3*sqrt(3)*(2*x - 1)) - 2*sqrt(3)*(d*x^3 - 2*e*x^2 - d*x - e))/(x^4 + x^2 + 1)

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Sympy [A]  time = 8.57345, size = 952, normalized size = 6.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-d/4 - sqrt(3)*I*(d + 2*e)/18)*log(x + (-10309*d**4*e + 1026*d**4*(-d/4 - sqrt(
3)*I*(d + 2*e)/18) - 7200*d**2*e**3 - 31536*d**2*e**2*(-d/4 - sqrt(3)*I*(d + 2*e
)/18) + 108432*d**2*e*(-d/4 - sqrt(3)*I*(d + 2*e)/18)**2 + 163296*d**2*(-d/4 - s
qrt(3)*I*(d + 2*e)/18)**3 + 1792*e**5 + 11520*e**4*(-d/4 - sqrt(3)*I*(d + 2*e)/1
8) + 48384*e**3*(-d/4 - sqrt(3)*I*(d + 2*e)/18)**2 + 311040*e**2*(-d/4 - sqrt(3)
*I*(d + 2*e)/18)**3)/(3348*d**5 - 11408*d**3*e**2 - 7936*d*e**4)) + (-d/4 + sqrt
(3)*I*(d + 2*e)/18)*log(x + (-10309*d**4*e + 1026*d**4*(-d/4 + sqrt(3)*I*(d + 2*
e)/18) - 7200*d**2*e**3 - 31536*d**2*e**2*(-d/4 + sqrt(3)*I*(d + 2*e)/18) + 1084
32*d**2*e*(-d/4 + sqrt(3)*I*(d + 2*e)/18)**2 + 163296*d**2*(-d/4 + sqrt(3)*I*(d
+ 2*e)/18)**3 + 1792*e**5 + 11520*e**4*(-d/4 + sqrt(3)*I*(d + 2*e)/18) + 48384*e
**3*(-d/4 + sqrt(3)*I*(d + 2*e)/18)**2 + 311040*e**2*(-d/4 + sqrt(3)*I*(d + 2*e)
/18)**3)/(3348*d**5 - 11408*d**3*e**2 - 7936*d*e**4)) + (d/4 - sqrt(3)*I*(d - 2*
e)/18)*log(x + (-10309*d**4*e + 1026*d**4*(d/4 - sqrt(3)*I*(d - 2*e)/18) - 7200*
d**2*e**3 - 31536*d**2*e**2*(d/4 - sqrt(3)*I*(d - 2*e)/18) + 108432*d**2*e*(d/4
- sqrt(3)*I*(d - 2*e)/18)**2 + 163296*d**2*(d/4 - sqrt(3)*I*(d - 2*e)/18)**3 + 1
792*e**5 + 11520*e**4*(d/4 - sqrt(3)*I*(d - 2*e)/18) + 48384*e**3*(d/4 - sqrt(3)
*I*(d - 2*e)/18)**2 + 311040*e**2*(d/4 - sqrt(3)*I*(d - 2*e)/18)**3)/(3348*d**5
- 11408*d**3*e**2 - 7936*d*e**4)) + (d/4 + sqrt(3)*I*(d - 2*e)/18)*log(x + (-103
09*d**4*e + 1026*d**4*(d/4 + sqrt(3)*I*(d - 2*e)/18) - 7200*d**2*e**3 - 31536*d*
*2*e**2*(d/4 + sqrt(3)*I*(d - 2*e)/18) + 108432*d**2*e*(d/4 + sqrt(3)*I*(d - 2*e
)/18)**2 + 163296*d**2*(d/4 + sqrt(3)*I*(d - 2*e)/18)**3 + 1792*e**5 + 11520*e**
4*(d/4 + sqrt(3)*I*(d - 2*e)/18) + 48384*e**3*(d/4 + sqrt(3)*I*(d - 2*e)/18)**2
+ 311040*e**2*(d/4 + sqrt(3)*I*(d - 2*e)/18)**3)/(3348*d**5 - 11408*d**3*e**2 -
7936*d*e**4)) - (d*x**3 - d*x - 2*e*x**2 - e)/(6*x**4 + 6*x**2 + 6)

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GIAC/XCAS [A]  time = 0.273584, size = 135, normalized size = 0.96 \[ \frac{1}{9} \, \sqrt{3}{\left (d - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{9} \, \sqrt{3}{\left (d + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \, d{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{4} \, d{\rm ln}\left (x^{2} - x + 1\right ) - \frac{d x^{3} - 2 \, x^{2} e - d x - e}{6 \,{\left (x^{4} + x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(3)*(d - 2*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/9*sqrt(3)*(d + 2*e)*arct
an(1/3*sqrt(3)*(2*x - 1)) + 1/4*d*ln(x^2 + x + 1) - 1/4*d*ln(x^2 - x + 1) - 1/6*
(d*x^3 - 2*x^2*e - d*x - e)/(x^4 + x^2 + 1)