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

Optimal. Leaf size=69 \[ \frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{6} \log \left (x^2+x+1\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.243243, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{6} \log \left (x^2+x+1\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.4503, size = 71, normalized size = 1.03 \[ x + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} - \frac{2 \operatorname{atanh}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x**2 - x + 1)/6 - log(x**2 + x + 1)/6 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/
3))/3 - sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/3 - 2*atanh(x)/3

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Mathematica [A]  time = 0.00817237, size = 78, normalized size = 1.13 \[ \frac{1}{6} \left (\log \left (x^2-x+1\right )-\log \left (x^2+x+1\right )+6 x+2 \log (1-x)-2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 67, normalized size = 1. \[ x-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ( 1+x \right ) }{3}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1/x^3+x^3)/(-1/x^3+x^3),x)

[Out]

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

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Maxima [A]  time = 0.86105, size = 89, normalized size = 1.29 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x - \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{3} \, \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.284883, size = 109, normalized size = 1.58 \[ \frac{1}{18} \, \sqrt{3}{\left (6 \, \sqrt{3} x - \sqrt{3} \log \left (x^{2} + x + 1\right ) + \sqrt{3} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3} \log \left (x + 1\right ) + 2 \, \sqrt{3} \log \left (x - 1\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.736112, size = 85, normalized size = 1.23 \[ x + \frac{\log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x - 1)/3 - log(x + 1)/3 + log(x**2 - x + 1)/6 - log(x**2 + x + 1)/6 - sq
rt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/3 - sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3
)/3

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GIAC/XCAS [A]  time = 0.26149, size = 92, normalized size = 1.33 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x - \frac{1}{6} \,{\rm ln}\left (x^{2} + x + 1\right ) + \frac{1}{6} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{1}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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