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

Optimal. Leaf size=38 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right )-\frac{\sqrt{x^2+x+1}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0131642, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {730, 724, 206} \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right )-\frac{\sqrt{x^2+x+1}}{x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[1 + x + x^2]),x]

[Out]

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

Rule 730

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e
^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{1+x+x^2}} \, dx &=-\frac{\sqrt{1+x+x^2}}{x}-\frac{1}{2} \int \frac{1}{x \sqrt{1+x+x^2}} \, dx\\ &=-\frac{\sqrt{1+x+x^2}}{x}+\operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{2+x}{\sqrt{1+x+x^2}}\right )\\ &=-\frac{\sqrt{1+x+x^2}}{x}+\frac{1}{2} \tanh ^{-1}\left (\frac{2+x}{2 \sqrt{1+x+x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.009296, size = 38, normalized size = 1. \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right )-\frac{\sqrt{x^2+x+1}}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 31, normalized size = 0.8 \begin{align*}{\frac{1}{2}{\it Artanh} \left ({\frac{2+x}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \right ) }-{\frac{1}{x}\sqrt{{x}^{2}+x+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^2+x+1)^(1/2),x)

[Out]

1/2*arctanh(1/2*(2+x)/(x^2+x+1)^(1/2))-(x^2+x+1)^(1/2)/x

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Maxima [A]  time = 1.45017, size = 50, normalized size = 1.32 \begin{align*} -\frac{\sqrt{x^{2} + x + 1}}{x} + \frac{1}{2} \, \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x \right |}} + \frac{2 \, \sqrt{3}}{3 \,{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^2+x+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.15017, size = 144, normalized size = 3.79 \begin{align*} \frac{x \log \left (-x + \sqrt{x^{2} + x + 1} + 1\right ) - x \log \left (-x + \sqrt{x^{2} + x + 1} - 1\right ) - 2 \, x - 2 \, \sqrt{x^{2} + x + 1}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^2+x+1)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**2+x+1)**(1/2),x)

[Out]

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

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Giac [B]  time = 1.08955, size = 90, normalized size = 2.37 \begin{align*} \frac{x - \sqrt{x^{2} + x + 1} + 2}{{\left (x - \sqrt{x^{2} + x + 1}\right )}^{2} - 1} + \frac{1}{2} \, \log \left ({\left | -x + \sqrt{x^{2} + x + 1} + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | -x + \sqrt{x^{2} + x + 1} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^2+x+1)^(1/2),x, algorithm="giac")

[Out]

(x - sqrt(x^2 + x + 1) + 2)/((x - sqrt(x^2 + x + 1))^2 - 1) + 1/2*log(abs(-x + sqrt(x^2 + x + 1) + 1)) - 1/2*l
og(abs(-x + sqrt(x^2 + x + 1) - 1))