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

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*Sqrt[1 + x + x^2])]/8

________________________________________________________________________________________

Rubi [A]  time = 0.0239868, 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, Rules used = {744, 806, 724, 206} \[ \frac{3 \sqrt{x^2+x+1}}{4 x}-\frac{\sqrt{x^2+x+1}}{2 x^2}+\frac{1}{8} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

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*Sqrt[1 + x + x^2])]/8

Rule 744

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[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(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] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(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, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[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^3 \sqrt{1+x+x^2}} \, dx &=-\frac{\sqrt{1+x+x^2}}{2 x^2}-\frac{1}{2} \int \frac{\frac{3}{2}+x}{x^2 \sqrt{1+x+x^2}} \, dx\\ &=-\frac{\sqrt{1+x+x^2}}{2 x^2}+\frac{3 \sqrt{1+x+x^2}}{4 x}-\frac{1}{8} \int \frac{1}{x \sqrt{1+x+x^2}} \, dx\\ &=-\frac{\sqrt{1+x+x^2}}{2 x^2}+\frac{3 \sqrt{1+x+x^2}}{4 x}+\frac{1}{4} \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}}{2 x^2}+\frac{3 \sqrt{1+x+x^2}}{4 x}+\frac{1}{8} \tanh ^{-1}\left (\frac{2+x}{2 \sqrt{1+x+x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0142266, size = 43, normalized size = 0.75 \[ \frac{1}{8} \left (\frac{2 \sqrt{x^2+x+1} (3 x-2)}{x^2}+\tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right )\right ) \]

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 + ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])])/8

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 44, normalized size = 0.8 \begin{align*}{\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}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.43785, size = 68, normalized size = 1.19 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^2+x+1)^(1/2),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))

________________________________________________________________________________________

Fricas [A]  time = 2.06931, size = 169, normalized size = 2.96 \begin{align*} \frac{x^{2} \log \left (-x + \sqrt{x^{2} + x + 1} + 1\right ) - x^{2} \log \left (-x + \sqrt{x^{2} + x + 1} - 1\right ) + 6 \, x^{2} + 2 \, \sqrt{x^{2} + x + 1}{\left (3 \, x - 2\right )}}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^2+x+1)^(1/2),x, algorithm="fricas")

[Out]

1/8*(x^2*log(-x + sqrt(x^2 + x + 1) + 1) - x^2*log(-x + sqrt(x^2 + x + 1) - 1) + 6*x^2 + 2*sqrt(x^2 + x + 1)*(
3*x - 2))/x^2

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [A]  time = 1.08011, size = 113, normalized size = 1.98 \begin{align*} \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} \, \log \left ({\left | -x + \sqrt{x^{2} + x + 1} + 1 \right |}\right ) - \frac{1}{8} \, \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^3/(x^2+x+1)^(1/2),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*log(ab
s(-x + sqrt(x^2 + x + 1) + 1)) - 1/8*log(abs(-x + sqrt(x^2 + x + 1) - 1))