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

Optimal. Leaf size=92 \[ -\frac{1}{4} d \log \left (x^2-x+1\right )+\frac{1}{4} d \log \left (x^2+x+1\right )-\frac{d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-(d*ArcTan[(1 - 2*x)/Sqrt[3]])/(2*Sqrt[3]) + (d*ArcTan[(1 + 2*x)/Sqrt[3]])/(2*Sq
rt[3]) + (e*ArcTan[(1 + 2*x^2)/Sqrt[3]])/Sqrt[3] - (d*Log[1 - x + x^2])/4 + (d*L
og[1 + x + x^2])/4

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Rubi [A]  time = 0.158581, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{4} d \log \left (x^2-x+1\right )+\frac{1}{4} d \log \left (x^2+x+1\right )-\frac{d \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{d \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{e \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*ArcTan[(1 - 2*x)/Sqrt[3]])/(2*Sqrt[3]) + (d*ArcTan[(1 + 2*x)/Sqrt[3]])/(2*Sq
rt[3]) + (e*ArcTan[(1 + 2*x^2)/Sqrt[3]])/Sqrt[3] - (d*Log[1 - x + x^2])/4 + (d*L
og[1 + x + x^2])/4

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Rubi in Sympy [A]  time = 21.704, size = 95, normalized size = 1.03 \[ - \frac{d \log{\left (x^{2} - x + 1 \right )}}{4} + \frac{d \log{\left (x^{2} + x + 1 \right )}}{4} + \frac{\sqrt{3} d \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} d \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} e \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(x**4+x**2+1),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))/6 + sqrt(3)*d*atan(sqrt(3)*(2*x/3 + 1/3))/6 + sqrt(3)*e*atan(sqrt(3)*(2*x
**2/3 + 1/3))/3

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Mathematica [C]  time = 0.325108, size = 98, normalized size = 1.07 \[ \frac{1}{6} i \left (\sqrt{6-6 i \sqrt{3}} d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )-\sqrt{6+6 i \sqrt{3}} d \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )+2 i \sqrt{3} e \tan ^{-1}\left (\frac{\sqrt{3}}{2 x^2+1}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.008, size = 92, normalized size = 1. \[{\frac{d\ln \left ({x}^{2}+x+1 \right ) }{4}}+{\frac{d\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}e}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{d\ln \left ({x}^{2}-x+1 \right ) }{4}}+{\frac{d\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}e}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(x^4+x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.782779, size = 88, normalized size = 0.96 \[ \frac{1}{6} \, \sqrt{3}{\left (d - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (d + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \, d \log \left (x^{2} + x + 1\right ) - \frac{1}{4} \, d \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*(d - 2*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*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)

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Fricas [A]  time = 0.267046, size = 93, normalized size = 1.01 \[ \frac{1}{12} \, \sqrt{3}{\left (\sqrt{3} d \log \left (x^{2} + x + 1\right ) - \sqrt{3} d \log \left (x^{2} - x + 1\right ) + 2 \,{\left (d - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left (d + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.45709, size = 923, normalized size = 10.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-d/4 - sqrt(3)*I*(d + 2*e)/12)*log(x + (-7*d**4*e + 6*d**4*(-d/4 - sqrt(3)*I*(d
+ 2*e)/12) - 15*d**2*e**3 - 18*d**2*e**2*(-d/4 - sqrt(3)*I*(d + 2*e)/12) + 60*d
**2*e*(-d/4 - sqrt(3)*I*(d + 2*e)/12)**2 + 72*d**2*(-d/4 - sqrt(3)*I*(d + 2*e)/1
2)**3 + 4*e**5 + 24*e**4*(-d/4 - sqrt(3)*I*(d + 2*e)/12) + 48*e**3*(-d/4 - sqrt(
3)*I*(d + 2*e)/12)**2 + 288*e**2*(-d/4 - sqrt(3)*I*(d + 2*e)/12)**3)/(3*d**5 - 8
*d**3*e**2 - 16*d*e**4)) + (-d/4 + sqrt(3)*I*(d + 2*e)/12)*log(x + (-7*d**4*e +
6*d**4*(-d/4 + sqrt(3)*I*(d + 2*e)/12) - 15*d**2*e**3 - 18*d**2*e**2*(-d/4 + sqr
t(3)*I*(d + 2*e)/12) + 60*d**2*e*(-d/4 + sqrt(3)*I*(d + 2*e)/12)**2 + 72*d**2*(-
d/4 + sqrt(3)*I*(d + 2*e)/12)**3 + 4*e**5 + 24*e**4*(-d/4 + sqrt(3)*I*(d + 2*e)/
12) + 48*e**3*(-d/4 + sqrt(3)*I*(d + 2*e)/12)**2 + 288*e**2*(-d/4 + sqrt(3)*I*(d
+ 2*e)/12)**3)/(3*d**5 - 8*d**3*e**2 - 16*d*e**4)) + (d/4 - sqrt(3)*I*(d - 2*e)
/12)*log(x + (-7*d**4*e + 6*d**4*(d/4 - sqrt(3)*I*(d - 2*e)/12) - 15*d**2*e**3 -
18*d**2*e**2*(d/4 - sqrt(3)*I*(d - 2*e)/12) + 60*d**2*e*(d/4 - sqrt(3)*I*(d - 2
*e)/12)**2 + 72*d**2*(d/4 - sqrt(3)*I*(d - 2*e)/12)**3 + 4*e**5 + 24*e**4*(d/4 -
sqrt(3)*I*(d - 2*e)/12) + 48*e**3*(d/4 - sqrt(3)*I*(d - 2*e)/12)**2 + 288*e**2*
(d/4 - sqrt(3)*I*(d - 2*e)/12)**3)/(3*d**5 - 8*d**3*e**2 - 16*d*e**4)) + (d/4 +
sqrt(3)*I*(d - 2*e)/12)*log(x + (-7*d**4*e + 6*d**4*(d/4 + sqrt(3)*I*(d - 2*e)/1
2) - 15*d**2*e**3 - 18*d**2*e**2*(d/4 + sqrt(3)*I*(d - 2*e)/12) + 60*d**2*e*(d/4
+ sqrt(3)*I*(d - 2*e)/12)**2 + 72*d**2*(d/4 + sqrt(3)*I*(d - 2*e)/12)**3 + 4*e*
*5 + 24*e**4*(d/4 + sqrt(3)*I*(d - 2*e)/12) + 48*e**3*(d/4 + sqrt(3)*I*(d - 2*e)
/12)**2 + 288*e**2*(d/4 + sqrt(3)*I*(d - 2*e)/12)**3)/(3*d**5 - 8*d**3*e**2 - 16
*d*e**4))

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GIAC/XCAS [A]  time = 0.281183, size = 90, normalized size = 0.98 \[ \frac{1}{6} \, \sqrt{3}{\left (d - 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (d + 2 \, e\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \, d{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{4} \, d{\rm ln}\left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*(d - 2*e)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*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)