3.2427 \(\int \frac{1}{x^3 \sqrt{1+x+x^2}} \, dx\)

Optimal. Leaf size=57 \[ \frac{3 \sqrt{x^2+x+1}}{4 x}-\frac{\sqrt{x^2+x+1}}{2 x^2}+\frac{1}{8} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

[Out]

-Sqrt[1 + x + x^2]/(2*x^2) + (3*Sqrt[1 + x + x^2])/(4*x) + ArcTanh[(2 + x)/(2*Sq
rt[1 + x + x^2])]/8

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Rubi [A]  time = 0.0843027, antiderivative size = 57, 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 \[ \frac{3 \sqrt{x^2+x+1}}{4 x}-\frac{\sqrt{x^2+x+1}}{2 x^2}+\frac{1}{8} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 + x + x^2]/(2*x^2) + (3*Sqrt[1 + x + x^2])/(4*x) + ArcTanh[(2 + x)/(2*Sq
rt[1 + x + x^2])]/8

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Rubi in Sympy [A]  time = 10.6358, size = 48, normalized size = 0.84 \[ \frac{\operatorname{atanh}{\left (\frac{x + 2}{2 \sqrt{x^{2} + x + 1}} \right )}}{8} + \frac{3 \sqrt{x^{2} + x + 1}}{4 x} - \frac{\sqrt{x^{2} + x + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh((x + 2)/(2*sqrt(x**2 + x + 1)))/8 + 3*sqrt(x**2 + x + 1)/(4*x) - sqrt(x**2
+ x + 1)/(2*x**2)

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Mathematica [A]  time = 0.0462935, size = 45, normalized size = 0.79 \[ \frac{1}{8} \left (\frac{2 \sqrt{x^2+x+1} (3 x-2)}{x^2}+\log \left (2 \sqrt{x^2+x+1}+x+2\right )-\log (x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*(-2 + 3*x)*Sqrt[1 + x + x^2])/x^2 - Log[x] + Log[2 + x + 2*Sqrt[1 + x + x^2]
])/8

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Maple [A]  time = 0.008, size = 44, normalized size = 0.8 \[{\frac{1}{8}{\it Artanh} \left ({\frac{2+x}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \right ) }-{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{2}+x+1}}+{\frac{3}{4\,x}\sqrt{{x}^{2}+x+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^2+x+1)^(1/2),x)

[Out]

1/8*arctanh(1/2*(2+x)/(x^2+x+1)^(1/2))-1/2*(x^2+x+1)^(1/2)/x^2+3/4*(x^2+x+1)^(1/
2)/x

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Maxima [A]  time = 0.745375, size = 68, normalized size = 1.19 \[ \frac{3 \, \sqrt{x^{2} + x + 1}}{4 \, x} - \frac{\sqrt{x^{2} + x + 1}}{2 \, x^{2}} + \frac{1}{8} \, \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x \right |}} + \frac{2 \, \sqrt{3}}{3 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*sqrt(x^2 + x + 1)/x - 1/2*sqrt(x^2 + x + 1)/x^2 + 1/8*arcsinh(1/3*sqrt(3)*x/
abs(x) + 2/3*sqrt(3)/abs(x))

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Fricas [A]  time = 0.220987, size = 234, normalized size = 4.11 \[ \frac{8 \, x^{3} + 6 \, x^{2} +{\left (8 \, x^{4} + 8 \, x^{3} + 5 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2}\right )} \sqrt{x^{2} + x + 1}\right )} \log \left (-x + \sqrt{x^{2} + x + 1} + 1\right ) -{\left (8 \, x^{4} + 8 \, x^{3} + 5 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2}\right )} \sqrt{x^{2} + x + 1}\right )} \log \left (-x + \sqrt{x^{2} + x + 1} - 1\right ) - 2 \,{\left (4 \, x^{2} + x + 10\right )} \sqrt{x^{2} + x + 1} + 24 \, x + 16}{8 \,{\left (8 \, x^{4} + 8 \, x^{3} + 5 \, x^{2} - 4 \,{\left (2 \, x^{3} + x^{2}\right )} \sqrt{x^{2} + x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*x^3 + 6*x^2 + (8*x^4 + 8*x^3 + 5*x^2 - 4*(2*x^3 + x^2)*sqrt(x^2 + x + 1))
*log(-x + sqrt(x^2 + x + 1) + 1) - (8*x^4 + 8*x^3 + 5*x^2 - 4*(2*x^3 + x^2)*sqrt
(x^2 + x + 1))*log(-x + sqrt(x^2 + x + 1) - 1) - 2*(4*x^2 + x + 10)*sqrt(x^2 + x
+ 1) + 24*x + 16)/(8*x^4 + 8*x^3 + 5*x^2 - 4*(2*x^3 + x^2)*sqrt(x^2 + x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211139, size = 113, normalized size = 1.98 \[ \frac{{\left (x - \sqrt{x^{2} + x + 1}\right )}^{3} + 9 \, x - 9 \, \sqrt{x^{2} + x + 1} + 8}{4 \,{\left ({\left (x - \sqrt{x^{2} + x + 1}\right )}^{2} - 1\right )}^{2}} + \frac{1}{8} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} + 1 \right |}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((x - sqrt(x^2 + x + 1))^3 + 9*x - 9*sqrt(x^2 + x + 1) + 8)/((x - sqrt(x^2 +
x + 1))^2 - 1)^2 + 1/8*ln(abs(-x + sqrt(x^2 + x + 1) + 1)) - 1/8*ln(abs(-x + sq
rt(x^2 + x + 1) - 1))