3.106 \(\int \frac{a+b x^2}{1+x^2+x^4} \, dx\)

Optimal. Leaf size=83 \[ -\frac{1}{4} (a-b) \log \left (x^2-x+1\right )+\frac{1}{4} (a-b) \log \left (x^2+x+1\right )-\frac{(a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{(a+b) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

-((a + b)*ArcTan[(1 - 2*x)/Sqrt[3]])/(2*Sqrt[3]) + ((a + b)*ArcTan[(1 + 2*x)/Sqr
t[3]])/(2*Sqrt[3]) - ((a - b)*Log[1 - x + x^2])/4 + ((a - b)*Log[1 + x + x^2])/4

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Rubi [A]  time = 0.124618, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{1}{4} (a-b) \log \left (x^2-x+1\right )+\frac{1}{4} (a-b) \log \left (x^2+x+1\right )-\frac{(a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{(a+b) \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(1 + x^2 + x^4),x]

[Out]

-((a + b)*ArcTan[(1 - 2*x)/Sqrt[3]])/(2*Sqrt[3]) + ((a + b)*ArcTan[(1 + 2*x)/Sqr
t[3]])/(2*Sqrt[3]) - ((a - b)*Log[1 - x + x^2])/4 + ((a - b)*Log[1 + x + x^2])/4

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Rubi in Sympy [A]  time = 19.396, size = 80, normalized size = 0.96 \[ - \left (\frac{a}{4} - \frac{b}{4}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{a}{4} - \frac{b}{4}\right ) \log{\left (x^{2} + x + 1 \right )} + \frac{\sqrt{3} \left (a + b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \left (a + b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/(x**4+x**2+1),x)

[Out]

-(a/4 - b/4)*log(x**2 - x + 1) + (a/4 - b/4)*log(x**2 + x + 1) + sqrt(3)*(a + b)
*atan(sqrt(3)*(2*x/3 - 1/3))/6 + sqrt(3)*(a + b)*atan(sqrt(3)*(2*x/3 + 1/3))/6

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Mathematica [C]  time = 0.209343, size = 97, normalized size = 1.17 \[ \frac{\left (2 i a+\left (\sqrt{3}-i\right ) b\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x\right )}{\sqrt{6+6 i \sqrt{3}}}+\frac{\left (\left (\sqrt{3}+i\right ) b-2 i a\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x\right )}{\sqrt{6-6 i \sqrt{3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^2)/(1 + x^2 + x^4),x]

[Out]

(((2*I)*a + (-I + Sqrt[3])*b)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[6 + (6*I)*Sqrt[
3]] + (((-2*I)*a + (I + Sqrt[3])*b)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[6 - (6*I)*
Sqrt[3]]

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Maple [A]  time = 0.007, size = 114, normalized size = 1.4 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) a}{4}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) b}{4}}+{\frac{\sqrt{3}a}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}b}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ({x}^{2}-x+1 \right ) a}{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) b}{4}}+{\frac{\sqrt{3}a}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}b}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(x^4+x^2+1),x)

[Out]

1/4*ln(x^2+x+1)*a-1/4*ln(x^2+x+1)*b+1/6*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*a+1/
6*3^(1/2)*arctan(1/3*(1+2*x)*3^(1/2))*b-1/4*ln(x^2-x+1)*a+1/4*ln(x^2-x+1)*b+1/6*
3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*a+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))*b

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Maxima [A]  time = 0.861314, size = 93, normalized size = 1.12 \[ \frac{1}{6} \, \sqrt{3}{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \,{\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac{1}{4} \,{\left (a - b\right )} \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*(a + b)*arctan(1
/3*sqrt(3)*(2*x - 1)) + 1/4*(a - b)*log(x^2 + x + 1) - 1/4*(a - b)*log(x^2 - x +
1)

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Fricas [A]  time = 0.294693, size = 99, normalized size = 1.19 \[ \frac{1}{12} \, \sqrt{3}{\left (\sqrt{3}{\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \sqrt{3}{\left (a - b\right )} \log \left (x^{2} - x + 1\right ) + 2 \,{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \,{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*(sqrt(3)*(a - b)*log(x^2 + x + 1) - sqrt(3)*(a - b)*log(x^2 - x + 1
) + 2*(a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 2*(a + b)*arctan(1/3*sqrt(3)*(2*x
- 1)))

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Sympy [A]  time = 2.95709, size = 740, normalized size = 8.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/(x**4+x**2+1),x)

[Out]

(-a/4 + b/4 - sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(-a/4 + b/4 - sqrt(3)*I*(a +
b)/12) + 6*a**2*b*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12) - 12*a*b**2*(-a/4 + b/4 -
sqrt(3)*I*(a + b)/12) + 24*a*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(-
a/4 + b/4 - sqrt(3)*I*(a + b)/12) - 48*b*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12)**3)
/(a**4 - a**3*b + a*b**3 - b**4)) + (-a/4 + b/4 + sqrt(3)*I*(a + b)/12)*log(x +
(2*a**3*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) + 6*a**2*b*(-a/4 + b/4 + sqrt(3)*I*(
a + b)/12) - 12*a*b**2*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) + 24*a*(-a/4 + b/4 +
sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) - 48*b*(-a
/4 + b/4 + sqrt(3)*I*(a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4)) + (a/4 - b
/4 - sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(a/4 - b/4 - sqrt(3)*I*(a + b)/12) +
6*a**2*b*(a/4 - b/4 - sqrt(3)*I*(a + b)/12) - 12*a*b**2*(a/4 - b/4 - sqrt(3)*I*(
a + b)/12) + 24*a*(a/4 - b/4 - sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(a/4 - b/4 - sq
rt(3)*I*(a + b)/12) - 48*b*(a/4 - b/4 - sqrt(3)*I*(a + b)/12)**3)/(a**4 - a**3*b
+ a*b**3 - b**4)) + (a/4 - b/4 + sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(a/4 - b
/4 + sqrt(3)*I*(a + b)/12) + 6*a**2*b*(a/4 - b/4 + sqrt(3)*I*(a + b)/12) - 12*a*
b**2*(a/4 - b/4 + sqrt(3)*I*(a + b)/12) + 24*a*(a/4 - b/4 + sqrt(3)*I*(a + b)/12
)**3 + 2*b**3*(a/4 - b/4 + sqrt(3)*I*(a + b)/12) - 48*b*(a/4 - b/4 + sqrt(3)*I*(
a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4))

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GIAC/XCAS [A]  time = 0.271006, size = 93, normalized size = 1.12 \[ \frac{1}{6} \, \sqrt{3}{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3}{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \,{\left (a - b\right )}{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{4} \,{\left (a - b\right )}{\rm ln}\left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*(a + b)*arctan(1
/3*sqrt(3)*(2*x - 1)) + 1/4*(a - b)*ln(x^2 + x + 1) - 1/4*(a - b)*ln(x^2 - x + 1
)