Integrand size = 14, antiderivative size = 57 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=-\frac {1}{7 x^7}+\frac {1}{5 x^5}-\frac {1}{x}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}} \] Output:
-1/7/x^7+1/5/x^5-1/x+1/3*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)-1/3*arctan(1/ 3*(1+2*x)*3^(1/2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=-\frac {1}{7 x^7}+\frac {1}{5 x^5}-\frac {1}{x}-\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {6+6 i \sqrt {3}}}-\frac {\left (-i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {6-6 i \sqrt {3}}} \] Input:
Integrate[1/(x^8*(1 + x^2 + x^4)),x]
Output:
-1/7*1/x^7 + 1/(5*x^5) - x^(-1) - ((I + Sqrt[3])*ArcTan[((-I + Sqrt[3])*x) /2])/Sqrt[6 + (6*I)*Sqrt[3]] - ((-I + Sqrt[3])*ArcTan[((I + Sqrt[3])*x)/2] )/Sqrt[6 - (6*I)*Sqrt[3]]
Time = 0.40 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {1443, 27, 1604, 27, 1443, 25, 1475, 1083, 217}
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.
\(\displaystyle \int \frac {1}{x^8 \left (x^4+x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 1443 |
\(\displaystyle \frac {1}{7} \int -\frac {7 \left (x^2+1\right )}{x^6 \left (x^4+x^2+1\right )}dx-\frac {1}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {x^2+1}{x^6 \left (x^4+x^2+1\right )}dx-\frac {1}{7 x^7}\) |
\(\Big \downarrow \) 1604 |
\(\displaystyle \frac {1}{5} \int \frac {5}{x^2 \left (x^4+x^2+1\right )}dx-\frac {1}{7 x^7}+\frac {1}{5 x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{x^2 \left (x^4+x^2+1\right )}dx-\frac {1}{7 x^7}+\frac {1}{5 x^5}\) |
\(\Big \downarrow \) 1443 |
\(\displaystyle \int -\frac {x^2+1}{x^4+x^2+1}dx-\frac {1}{7 x^7}+\frac {1}{5 x^5}-\frac {1}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {x^2+1}{x^4+x^2+1}dx-\frac {1}{7 x^7}+\frac {1}{5 x^5}-\frac {1}{x}\) |
\(\Big \downarrow \) 1475 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1}{x^2+x+1}dx-\frac {1}{7 x^7}+\frac {1}{5 x^5}-\frac {1}{x}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)+\int \frac {1}{-(2 x+1)^2-3}d(2 x+1)-\frac {1}{7 x^7}+\frac {1}{5 x^5}-\frac {1}{x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {\arctan \left (\frac {2 x-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{7 x^7}+\frac {1}{5 x^5}-\frac {1}{x}\) |
Input:
Int[1/(x^8*(1 + x^2 + x^4)),x]
Output:
-1/7*1/x^7 + 1/(5*x^5) - x^(-1) - ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3] - Arc Tan[(1 + 2*x)/Sqrt[3]]/Sqrt[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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim p[1/(a*d^2*(m + 1)) Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^ 2, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) /(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}-\frac {1}{7 x^{7}}-\frac {1}{x}+\frac {1}{5 x^{5}}\) | \(49\) |
risch | \(\frac {-x^{6}+\frac {1}{5} x^{2}-\frac {1}{7}}{x^{7}}-\frac {\sqrt {3}\, \arctan \left (\frac {x^{3} \sqrt {3}}{3}+\frac {2 \sqrt {3}\, x}{3}\right )}{3}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{3}\right )}{3}\) | \(51\) |
Input:
int(1/x^8/(x^4+x^2+1),x,method=_RETURNVERBOSE)
Output:
-1/3*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/3*3^(1/2)*arctan(1/3*(2*x-1)*3^ (1/2))-1/7/x^7-1/x+1/5/x^5
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=-\frac {35 \, \sqrt {3} x^{7} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x^{3} + 2 \, x\right )}\right ) + 35 \, \sqrt {3} x^{7} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + 105 \, x^{6} - 21 \, x^{2} + 15}{105 \, x^{7}} \] Input:
integrate(1/x^8/(x^4+x^2+1),x, algorithm="fricas")
Output:
-1/105*(35*sqrt(3)*x^7*arctan(1/3*sqrt(3)*(x^3 + 2*x)) + 35*sqrt(3)*x^7*ar ctan(1/3*sqrt(3)*x) + 105*x^6 - 21*x^2 + 15)/x^7
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=\frac {\sqrt {3} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{3}}{3} + \frac {2 \sqrt {3} x}{3} \right )}\right )}{6} + \frac {- 35 x^{6} + 7 x^{2} - 5}{35 x^{7}} \] Input:
integrate(1/x**8/(x**4+x**2+1),x)
Output:
sqrt(3)*(-2*atan(sqrt(3)*x/3) - 2*atan(sqrt(3)*x**3/3 + 2*sqrt(3)*x/3))/6 + (-35*x**6 + 7*x**2 - 5)/(35*x**7)
Time = 0.13 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {35 \, x^{6} - 7 \, x^{2} + 5}{35 \, x^{7}} \] Input:
integrate(1/x^8/(x^4+x^2+1),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/3*sqrt(3)*arctan(1/3*sqrt(3 )*(2*x - 1)) - 1/35*(35*x^6 - 7*x^2 + 5)/x^7
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {35 \, x^{6} - 7 \, x^{2} + 5}{35 \, x^{7}} \] Input:
integrate(1/x^8/(x^4+x^2+1),x, algorithm="giac")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/3*sqrt(3)*arctan(1/3*sqrt(3 )*(2*x - 1)) - 1/35*(35*x^6 - 7*x^2 + 5)/x^7
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=-\frac {\sqrt {3}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x^3}{3}+\frac {2\,\sqrt {3}\,x}{3}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}\right )\right )}{6}-\frac {x^6-\frac {x^2}{5}+\frac {1}{7}}{x^7} \] Input:
int(1/(x^8*(x^2 + x^4 + 1)),x)
Output:
- (3^(1/2)*(2*atan((2*3^(1/2)*x)/3 + (3^(1/2)*x^3)/3) + 2*atan((3^(1/2)*x) /3)))/6 - (x^6 - x^2/5 + 1/7)/x^7
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^8 \left (1+x^2+x^4\right )} \, dx=\frac {-35 \sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right ) x^{7}-35 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{7}-105 x^{6}+21 x^{2}-15}{105 x^{7}} \] Input:
int(1/x^8/(x^4+x^2+1),x)
Output:
( - 35*sqrt(3)*atan((2*x - 1)/sqrt(3))*x**7 - 35*sqrt(3)*atan((2*x + 1)/sq rt(3))*x**7 - 105*x**6 + 21*x**2 - 15)/(105*x**7)