Integrand size = 14, antiderivative size = 62 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {2 (1-x)}{3 x \sqrt {1+x+x^2}}-\frac {5 \sqrt {1+x+x^2}}{3 x}+\frac {3}{2} \text {arctanh}\left (\frac {2+x}{2 \sqrt {1+x+x^2}}\right ) \] Output:
3/2*arctanh(1/2*(2+x)/(x^2+x+1)^(1/2))+2/3*(1-x)/x/(x^2+x+1)^(1/2)-5/3*(x^ 2+x+1)^(1/2)/x
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {-3-7 x-5 x^2}{3 x \sqrt {1+x+x^2}}-3 \text {arctanh}\left (x-\sqrt {1+x+x^2}\right ) \] Input:
Integrate[1/(x^2*(1 + x + x^2)^(3/2)),x]
Output:
(-3 - 7*x - 5*x^2)/(3*x*Sqrt[1 + x + x^2]) - 3*ArcTanh[x - Sqrt[1 + x + x^ 2]]
Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1165, 27, 1228, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^2 \left (x^2+x+1\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1165 |
\(\displaystyle \frac {2}{3} \int \frac {5-2 x}{2 x^2 \sqrt {x^2+x+1}}dx+\frac {2 (1-x)}{3 x \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {5-2 x}{x^2 \sqrt {x^2+x+1}}dx+\frac {2 (1-x)}{3 x \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {1}{3} \left (-\frac {9}{2} \int \frac {1}{x \sqrt {x^2+x+1}}dx-\frac {5 \sqrt {x^2+x+1}}{x}\right )+\frac {2 (1-x)}{3 x \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {1}{3} \left (9 \int \frac {1}{4-\frac {(x+2)^2}{x^2+x+1}}d\frac {x+2}{\sqrt {x^2+x+1}}-\frac {5 \sqrt {x^2+x+1}}{x}\right )+\frac {2 (1-x)}{3 x \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\frac {9}{2} \text {arctanh}\left (\frac {x+2}{2 \sqrt {x^2+x+1}}\right )-\frac {5 \sqrt {x^2+x+1}}{x}\right )+\frac {2 (1-x)}{3 x \sqrt {x^2+x+1}}\) |
Input:
Int[1/(x^2*(1 + x + x^2)^(3/2)),x]
Output:
(2*(1 - x))/(3*x*Sqrt[1 + x + x^2]) + ((-5*Sqrt[1 + x + x^2])/x + (9*ArcTa nh[(2 + x)/(2*Sqrt[1 + x + x^2])])/2)/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[(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 ] && EqQ[Simplify[m + 2*p + 3], 0]
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {5 x^{2}+7 x +3}{3 \sqrt {x^{2}+x +1}\, x}+\frac {3 \,\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{2}\) | \(41\) |
trager | \(-\frac {5 x^{2}+7 x +3}{3 \sqrt {x^{2}+x +1}\, x}+\frac {3 \ln \left (\frac {2 \sqrt {x^{2}+x +1}+2+x}{x}\right )}{2}\) | \(45\) |
default | \(-\frac {1}{x \sqrt {x^{2}+x +1}}-\frac {3}{2 \sqrt {x^{2}+x +1}}-\frac {5 \left (1+2 x \right )}{6 \sqrt {x^{2}+x +1}}+\frac {3 \,\operatorname {arctanh}\left (\frac {2+x}{2 \sqrt {x^{2}+x +1}}\right )}{2}\) | \(56\) |
Input:
int(1/x^2/(x^2+x+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(5*x^2+7*x+3)/(x^2+x+1)^(1/2)/x+3/2*arctanh(1/2*(2+x)/(x^2+x+1)^(1/2) )
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (46) = 92\).
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.52 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=-\frac {10 \, x^{3} + 10 \, x^{2} - 9 \, {\left (x^{3} + x^{2} + x\right )} \log \left (-x + \sqrt {x^{2} + x + 1} + 1\right ) + 9 \, {\left (x^{3} + x^{2} + x\right )} \log \left (-x + \sqrt {x^{2} + x + 1} - 1\right ) + 2 \, {\left (5 \, x^{2} + 7 \, x + 3\right )} \sqrt {x^{2} + x + 1} + 10 \, x}{6 \, {\left (x^{3} + x^{2} + x\right )}} \] Input:
integrate(1/x^2/(x^2+x+1)^(3/2),x, algorithm="fricas")
Output:
-1/6*(10*x^3 + 10*x^2 - 9*(x^3 + x^2 + x)*log(-x + sqrt(x^2 + x + 1) + 1) + 9*(x^3 + x^2 + x)*log(-x + sqrt(x^2 + x + 1) - 1) + 2*(5*x^2 + 7*x + 3)* sqrt(x^2 + x + 1) + 10*x)/(x^3 + x^2 + x)
\[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (x^{2} + x + 1\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**2/(x**2+x+1)**(3/2),x)
Output:
Integral(1/(x**2*(x**2 + x + 1)**(3/2)), x)
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=-\frac {5 \, x}{3 \, \sqrt {x^{2} + x + 1}} - \frac {7}{3 \, \sqrt {x^{2} + x + 1}} - \frac {1}{\sqrt {x^{2} + x + 1} x} + \frac {3}{2} \, \operatorname {arsinh}\left (\frac {\sqrt {3} x}{3 \, {\left | x \right |}} + \frac {2 \, \sqrt {3}}{3 \, {\left | x \right |}}\right ) \] Input:
integrate(1/x^2/(x^2+x+1)^(3/2),x, algorithm="maxima")
Output:
-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))
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (x + 2\right )}}{3 \, \sqrt {x^{2} + x + 1}} + \frac {x - \sqrt {x^{2} + x + 1} + 2}{{\left (x - \sqrt {x^{2} + x + 1}\right )}^{2} - 1} + \frac {3}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} + 1 \right |}\right ) - \frac {3}{2} \, \log \left ({\left | -x + \sqrt {x^{2} + x + 1} - 1 \right |}\right ) \] Input:
integrate(1/x^2/(x^2+x+1)^(3/2),x, algorithm="giac")
Output:
-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*log(abs(-x + sqrt(x^2 + x + 1) + 1)) - 3/2*log(ab s(-x + sqrt(x^2 + x + 1) - 1))
Timed out. \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (x^2+x+1\right )}^{3/2}} \,d x \] Input:
int(1/(x^2*(x + x^2 + 1)^(3/2)),x)
Output:
int(1/(x^2*(x + x^2 + 1)^(3/2)), x)
Time = 0.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.97 \[ \int \frac {1}{x^2 \left (1+x+x^2\right )^{3/2}} \, dx=\frac {-10 \sqrt {x^{2}+x +1}\, x^{2}-14 \sqrt {x^{2}+x +1}\, x -6 \sqrt {x^{2}+x +1}+9 \,\mathrm {log}\left (-2 \sqrt {x^{2}+x +1}-x -2\right ) x^{3}+9 \,\mathrm {log}\left (-2 \sqrt {x^{2}+x +1}-x -2\right ) x^{2}+9 \,\mathrm {log}\left (-2 \sqrt {x^{2}+x +1}-x -2\right ) x -9 \,\mathrm {log}\left (x \right ) x^{3}-9 \,\mathrm {log}\left (x \right ) x^{2}-9 \,\mathrm {log}\left (x \right ) x}{6 x \left (x^{2}+x +1\right )} \] Input:
int(1/x^2/(x^2+x+1)^(3/2),x)
Output:
( - 10*sqrt(x**2 + x + 1)*x**2 - 14*sqrt(x**2 + x + 1)*x - 6*sqrt(x**2 + x + 1) + 9*log( - 2*sqrt(x**2 + x + 1) - x - 2)*x**3 + 9*log( - 2*sqrt(x**2 + x + 1) - x - 2)*x**2 + 9*log( - 2*sqrt(x**2 + x + 1) - x - 2)*x - 9*log (x)*x**3 - 9*log(x)*x**2 - 9*log(x)*x)/(6*x*(x**2 + x + 1))