3.2297 \(\int \frac{a+b x}{(1+x) \left (1-x+x^2\right )} \, dx\)

Optimal. Leaf size=54 \[ -\frac{1}{6} (a-b) \log \left (x^2-x+1\right )+\frac{1}{3} (a-b) \log (x+1)-\frac{(a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-(((a + b)*ArcTan[(1 - 2*x)/Sqrt[3]])/Sqrt[3]) + ((a - b)*Log[1 + x])/3 - ((a -
b)*Log[1 - x + x^2])/6

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Rubi [A]  time = 0.138548, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{1}{6} (a-b) \log \left (x^2-x+1\right )+\frac{1}{3} (a-b) \log (x+1)-\frac{(a+b) \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(((a + b)*ArcTan[(1 - 2*x)/Sqrt[3]])/Sqrt[3]) + ((a - b)*Log[1 + x])/3 - ((a -
b)*Log[1 - x + x^2])/6

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Rubi in Sympy [A]  time = 16.7299, size = 51, normalized size = 0.94 \[ - \left (\frac{a}{6} - \frac{b}{6}\right ) \log{\left (x^{2} - x + 1 \right )} + \left (\frac{a}{3} - \frac{b}{3}\right ) \log{\left (x + 1 \right )} + \frac{\sqrt{3} \left (a + b\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0477274, size = 49, normalized size = 0.91 \[ \frac{1}{6} (a-b) \left (2 \log (x+1)-\log \left (x^2-x+1\right )\right )+\frac{(a+b) \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b)*ArcTan[(-1 + 2*x)/Sqrt[3]])/Sqrt[3] + ((a - b)*(2*Log[1 + x] - Log[1 -
x + x^2]))/6

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Maple [A]  time = 0.044, size = 74, normalized size = 1.4 \[ -{\frac{\ln \left ({x}^{2}-x+1 \right ) a}{6}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) b}{6}}+{\frac{\sqrt{3}a}{3}\arctan \left ({\frac{ \left ( -1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{b\sqrt{3}}{3}\arctan \left ({\frac{ \left ( -1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+x \right ) a}{3}}-{\frac{\ln \left ( 1+x \right ) b}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(1+x)/(x^2-x+1),x)

[Out]

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

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Maxima [A]  time = 0.767058, size = 63, normalized size = 1.17 \[ \frac{1}{3} \, \sqrt{3}{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{6} \,{\left (a - b\right )} \log \left (x^{2} - x + 1\right ) + \frac{1}{3} \,{\left (a - b\right )} \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27176, size = 73, normalized size = 1.35 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3}{\left (a - b\right )} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3}{\left (a - b\right )} \log \left (x + 1\right ) - 6 \,{\left (a + b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*(a - b)*log(x^2 - x + 1) - 2*sqrt(3)*(a - b)*log(x + 1) -
6*(a + b)*arctan(1/3*sqrt(3)*(2*x - 1)))

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Sympy [A]  time = 1.16302, size = 201, normalized size = 3.72 \[ \frac{\left (a - b\right ) \log{\left (x + \frac{a^{2} \left (a - b\right ) + 2 a b^{2} + b \left (a - b\right )^{2}}{a^{3} + b^{3}} \right )}}{3} + \left (- \frac{a}{6} + \frac{b}{6} - \frac{\sqrt{3} i \left (a + b\right )}{6}\right ) \log{\left (x + \frac{3 a^{2} \left (- \frac{a}{6} + \frac{b}{6} - \frac{\sqrt{3} i \left (a + b\right )}{6}\right ) + 2 a b^{2} + 9 b \left (- \frac{a}{6} + \frac{b}{6} - \frac{\sqrt{3} i \left (a + b\right )}{6}\right )^{2}}{a^{3} + b^{3}} \right )} + \left (- \frac{a}{6} + \frac{b}{6} + \frac{\sqrt{3} i \left (a + b\right )}{6}\right ) \log{\left (x + \frac{3 a^{2} \left (- \frac{a}{6} + \frac{b}{6} + \frac{\sqrt{3} i \left (a + b\right )}{6}\right ) + 2 a b^{2} + 9 b \left (- \frac{a}{6} + \frac{b}{6} + \frac{\sqrt{3} i \left (a + b\right )}{6}\right )^{2}}{a^{3} + b^{3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a - b)*log(x + (a**2*(a - b) + 2*a*b**2 + b*(a - b)**2)/(a**3 + b**3))/3 + (-a/
6 + b/6 - sqrt(3)*I*(a + b)/6)*log(x + (3*a**2*(-a/6 + b/6 - sqrt(3)*I*(a + b)/6
) + 2*a*b**2 + 9*b*(-a/6 + b/6 - sqrt(3)*I*(a + b)/6)**2)/(a**3 + b**3)) + (-a/6
+ b/6 + sqrt(3)*I*(a + b)/6)*log(x + (3*a**2*(-a/6 + b/6 + sqrt(3)*I*(a + b)/6)
+ 2*a*b**2 + 9*b*(-a/6 + b/6 + sqrt(3)*I*(a + b)/6)**2)/(a**3 + b**3))

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GIAC/XCAS [A]  time = 0.260096, size = 72, normalized size = 1.33 \[ \frac{1}{3} \,{\left (\sqrt{3} a + \sqrt{3} b\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{6} \,{\left (a - b\right )}{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{3} \,{\left (a - b\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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