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

Optimal. Leaf size=62 \[ \frac{2 (1-x)}{3 x \sqrt{x^2+x+1}}-\frac{5 \sqrt{x^2+x+1}}{3 x}+\frac{3}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

[Out]

(2*(1 - x))/(3*x*Sqrt[1 + x + x^2]) - (5*Sqrt[1 + x + x^2])/(3*x) + (3*ArcTanh[(
2 + x)/(2*Sqrt[1 + x + x^2])])/2

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Rubi [A]  time = 0.0756296, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{2 (1-x)}{3 x \sqrt{x^2+x+1}}-\frac{5 \sqrt{x^2+x+1}}{3 x}+\frac{3}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(1 - x))/(3*x*Sqrt[1 + x + x^2]) - (5*Sqrt[1 + x + x^2])/(3*x) + (3*ArcTanh[(
2 + x)/(2*Sqrt[1 + x + x^2])])/2

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Rubi in Sympy [A]  time = 4.94227, size = 53, normalized size = 0.85 \[ \frac{3 \operatorname{atanh}{\left (\frac{x + 2}{2 \sqrt{x^{2} + x + 1}} \right )}}{2} + \frac{2 \left (- x + 1\right )}{3 x \sqrt{x^{2} + x + 1}} - \frac{5 \sqrt{x^{2} + x + 1}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*atanh((x + 2)/(2*sqrt(x**2 + x + 1)))/2 + 2*(-x + 1)/(3*x*sqrt(x**2 + x + 1))
- 5*sqrt(x**2 + x + 1)/(3*x)

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Mathematica [A]  time = 0.0519758, size = 52, normalized size = 0.84 \[ \frac{1}{6} \left (-\frac{2 \left (5 x^2+7 x+3\right )}{x \sqrt{x^2+x+1}}+9 \log \left (2 \sqrt{x^2+x+1}+x+2\right )-9 \log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*(3 + 7*x + 5*x^2))/(x*Sqrt[1 + x + x^2]) - 9*Log[x] + 9*Log[2 + x + 2*Sqrt[
1 + x + x^2]])/6

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \[ -{\frac{1}{x}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}-{\frac{3}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}-{\frac{5+10\,x}{6}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{3}{2}{\it Artanh} \left ({\frac{2+x}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^2+x+1)^(3/2),x)

[Out]

-1/x/(x^2+x+1)^(1/2)-3/2/(x^2+x+1)^(1/2)-5/6*(1+2*x)/(x^2+x+1)^(1/2)+3/2*arctanh
(1/2*(2+x)/(x^2+x+1)^(1/2))

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Maxima [A]  time = 1.57382, size = 78, normalized size = 1.26 \[ -\frac{5 \, x}{3 \, \sqrt{x^{2} + x + 1}} - \frac{7}{3 \, \sqrt{x^{2} + x + 1}} - \frac{1}{\sqrt{x^{2} + x + 1} x} + \frac{3}{2} \, \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x \right |}} + \frac{2 \, \sqrt{3}}{3 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/3*x/sqrt(x^2 + x + 1) - 7/3/sqrt(x^2 + x + 1) - 1/(sqrt(x^2 + x + 1)*x) + 3/2
*arcsinh(1/3*sqrt(3)*x/abs(x) + 2/3*sqrt(3)/abs(x))

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Fricas [A]  time = 0.209398, size = 270, normalized size = 4.35 \[ \frac{24 \, x^{3} + 30 \, x^{2} + 3 \,{\left (8 \, x^{4} + 12 \, x^{3} + 12 \, x^{2} -{\left (8 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \sqrt{x^{2} + x + 1} + 4 \, x\right )} \log \left (-x + \sqrt{x^{2} + x + 1} + 1\right ) - 3 \,{\left (8 \, x^{4} + 12 \, x^{3} + 12 \, x^{2} -{\left (8 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \sqrt{x^{2} + x + 1} + 4 \, x\right )} \log \left (-x + \sqrt{x^{2} + x + 1} - 1\right ) - 2 \,{\left (12 \, x^{2} + 9 \, x + 4\right )} \sqrt{x^{2} + x + 1} + 26 \, x + 10}{2 \,{\left (8 \, x^{4} + 12 \, x^{3} + 12 \, x^{2} -{\left (8 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \sqrt{x^{2} + x + 1} + 4 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(24*x^3 + 30*x^2 + 3*(8*x^4 + 12*x^3 + 12*x^2 - (8*x^3 + 8*x^2 + 5*x)*sqrt(x
^2 + x + 1) + 4*x)*log(-x + sqrt(x^2 + x + 1) + 1) - 3*(8*x^4 + 12*x^3 + 12*x^2
- (8*x^3 + 8*x^2 + 5*x)*sqrt(x^2 + x + 1) + 4*x)*log(-x + sqrt(x^2 + x + 1) - 1)
- 2*(12*x^2 + 9*x + 4)*sqrt(x^2 + x + 1) + 26*x + 10)/(8*x^4 + 12*x^3 + 12*x^2
- (8*x^3 + 8*x^2 + 5*x)*sqrt(x^2 + x + 1) + 4*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (x^{2} + x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(x**2 + x + 1)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.204623, size = 108, normalized size = 1.74 \[ -\frac{2 \,{\left (x + 2\right )}}{3 \, \sqrt{x^{2} + x + 1}} + \frac{x - \sqrt{x^{2} + x + 1} + 2}{{\left (x - \sqrt{x^{2} + x + 1}\right )}^{2} - 1} + \frac{3}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} + 1 \right |}\right ) - \frac{3}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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