3.78 \(\int \frac{1-x+4 x^3}{1+x^3} \, dx\)

Optimal. Leaf size=44 \[ \frac{1}{3} \log \left (x^2-x+1\right )+4 x-\frac{2}{3} \log (x+1)+\frac{4 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0752904, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{1}{3} \log \left (x^2-x+1\right )+4 x-\frac{2}{3} \log (x+1)+\frac{4 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x + 4*x^3)/(1 + x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.5135, size = 44, normalized size = 1. \[ 4 x - \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{3} - \frac{4 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**3-x+1)/(x**3+1),x)

[Out]

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

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Mathematica [A]  time = 0.0145698, size = 44, normalized size = 1. \[ \frac{1}{3} \log \left (x^2-x+1\right )+4 x-\frac{2}{3} \log (x+1)-\frac{4 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x + 4*x^3)/(1 + x^3),x]

[Out]

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

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Maple [A]  time = 0.008, size = 38, normalized size = 0.9 \[ 4\,x+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}}-{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\ln \left ( 1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^3-x+1)/(x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.52395, size = 50, normalized size = 1.14 \[ -\frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 4 \, x + \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{2}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - x + 1)/(x^3 + 1),x, algorithm="maxima")

[Out]

-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 4*x + 1/3*log(x^2 - x + 1) - 2/3*lo
g(x + 1)

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Fricas [A]  time = 0.233204, size = 63, normalized size = 1.43 \[ \frac{1}{9} \, \sqrt{3}{\left (12 \, \sqrt{3} x + \sqrt{3} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3} \log \left (x + 1\right ) - 12 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - x + 1)/(x^3 + 1),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*(12*sqrt(3)*x + sqrt(3)*log(x^2 - x + 1) - 2*sqrt(3)*log(x + 1) - 12
*arctan(1/3*sqrt(3)*(2*x - 1)))

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Sympy [A]  time = 0.15081, size = 48, normalized size = 1.09 \[ 4 x - \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{3} - \frac{4 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**3-x+1)/(x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212904, size = 51, normalized size = 1.16 \[ -\frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 4 \, x + \frac{1}{3} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - x + 1)/(x^3 + 1),x, algorithm="giac")

[Out]

-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 4*x + 1/3*ln(x^2 - x + 1) - 2/3*ln(
abs(x + 1))