3.181 \(\int \frac{1+x^2}{x \left (1+x^3\right )^2} \, dx\)

Optimal. Leaf size=64 \[ -\frac{5}{18} \log \left (x^2-x+1\right )+\frac{x \left (x-x^2\right )}{3 \left (x^3+1\right )}+\log (x)-\frac{4}{9} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(x*(x - x^2))/(3*(1 + x^3)) - ArcTan[(1 - 2*x)/Sqrt[3]]/(3*Sqrt[3]) + Log[x] - (
4*Log[1 + x])/9 - (5*Log[1 - x + x^2])/18

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Rubi [A]  time = 0.12952, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{5}{18} \log \left (x^2-x+1\right )+\frac{x \left (x-x^2\right )}{3 \left (x^3+1\right )}+\log (x)-\frac{4}{9} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^2)/(x*(1 + x^3)^2),x]

[Out]

(x*(x - x^2))/(3*(1 + x^3)) - ArcTan[(1 - 2*x)/Sqrt[3]]/(3*Sqrt[3]) + Log[x] - (
4*Log[1 + x])/9 - (5*Log[1 - x + x^2])/18

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Rubi in Sympy [A]  time = 6.12965, size = 51, normalized size = 0.8 \[ \frac{x \left (x + \frac{1}{x}\right )}{3 \left (x^{3} + 1\right )} - \frac{\log{\left (x + 1 \right )}}{9} + \frac{\log{\left (x^{2} - x + 1 \right )}}{18} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+1)/x/(x**3+1)**2,x)

[Out]

x*(x + 1/x)/(3*(x**3 + 1)) - log(x + 1)/9 + log(x**2 - x + 1)/18 + sqrt(3)*atan(
sqrt(3)*(2*x/3 - 1/3))/9

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Mathematica [A]  time = 0.0515672, size = 65, normalized size = 1.02 \[ \frac{1}{18} \left (-6 \log \left (x^3+1\right )+\log \left (x^2-x+1\right )+\frac{6 \left (x^2+1\right )}{x^3+1}+18 \log (x)-2 \log (x+1)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^2)/(x*(1 + x^3)^2),x]

[Out]

((6*(1 + x^2))/(1 + x^3) + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 18*Log[x] - 2*
Log[1 + x] + Log[1 - x + x^2] - 6*Log[1 + x^3])/18

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Maple [A]  time = 0.018, size = 61, normalized size = 1. \[{\frac{2}{9+9\,x}}-{\frac{4\,\ln \left ( 1+x \right ) }{9}}+\ln \left ( x \right ) -{\frac{-1-x}{9\,{x}^{2}-9\,x+9}}-{\frac{5\,\ln \left ({x}^{2}-x+1 \right ) }{18}}+{\frac{\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)/x/(x^3+1)^2,x)

[Out]

2/9/(1+x)-4/9*ln(1+x)+ln(x)-1/9*(-1-x)/(x^2-x+1)-5/18*ln(x^2-x+1)+1/9*3^(1/2)*ar
ctan(1/3*(2*x-1)*3^(1/2))

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Maxima [A]  time = 1.59685, size = 68, normalized size = 1.06 \[ \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{x^{2} + 1}{3 \,{\left (x^{3} + 1\right )}} - \frac{5}{18} \, \log \left (x^{2} - x + 1\right ) - \frac{4}{9} \, \log \left (x + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/((x^3 + 1)^2*x),x, algorithm="maxima")

[Out]

1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/3*(x^2 + 1)/(x^3 + 1) - 5/18*log(x
^2 - x + 1) - 4/9*log(x + 1) + log(x)

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Fricas [A]  time = 0.2285, size = 116, normalized size = 1.81 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (x^{3} + 1\right )} \log \left (x^{2} - x + 1\right ) + 8 \, \sqrt{3}{\left (x^{3} + 1\right )} \log \left (x + 1\right ) - 18 \, \sqrt{3}{\left (x^{3} + 1\right )} \log \left (x\right ) - 6 \,{\left (x^{3} + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 6 \, \sqrt{3}{\left (x^{2} + 1\right )}\right )}}{54 \,{\left (x^{3} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/((x^3 + 1)^2*x),x, algorithm="fricas")

[Out]

-1/54*sqrt(3)*(5*sqrt(3)*(x^3 + 1)*log(x^2 - x + 1) + 8*sqrt(3)*(x^3 + 1)*log(x
+ 1) - 18*sqrt(3)*(x^3 + 1)*log(x) - 6*(x^3 + 1)*arctan(1/3*sqrt(3)*(2*x - 1)) -
6*sqrt(3)*(x^2 + 1))/(x^3 + 1)

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Sympy [A]  time = 0.318244, size = 60, normalized size = 0.94 \[ \frac{x^{2} + 1}{3 x^{3} + 3} + \log{\left (x \right )} - \frac{4 \log{\left (x + 1 \right )}}{9} - \frac{5 \log{\left (x^{2} - x + 1 \right )}}{18} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)/x/(x**3+1)**2,x)

[Out]

(x**2 + 1)/(3*x**3 + 3) + log(x) - 4*log(x + 1)/9 - 5*log(x**2 - x + 1)/18 + sqr
t(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/9

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GIAC/XCAS [A]  time = 0.216559, size = 81, normalized size = 1.27 \[ \frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{x^{2} + 1}{3 \,{\left (x^{2} - x + 1\right )}{\left (x + 1\right )}} - \frac{5}{18} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{4}{9} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)/((x^3 + 1)^2*x),x, algorithm="giac")

[Out]

1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/3*(x^2 + 1)/((x^2 - x + 1)*(x + 1)
) - 5/18*ln(x^2 - x + 1) - 4/9*ln(abs(x + 1)) + ln(abs(x))