Integrand size = 16, antiderivative size = 77 \[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}-\frac {1}{2} \arctan \left (\sqrt {3}-2 x\right )+\frac {1}{2} \arctan \left (\sqrt {3}+2 x\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} x}{1+x^2}\right )}{2 \sqrt {3}} \] Output:
-1/9/x^9-1/7/x^7+1/3/x^3+1/x+1/2*arctan(-3^(1/2)+2*x)+1/2*arctan(3^(1/2)+2 *x)-1/6*arctanh(3^(1/2)*x/(x^2+1))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}+\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )}{\sqrt {-6+6 i \sqrt {3}}}+\frac {\left (-i+\sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )}{\sqrt {-6-6 i \sqrt {3}}} \] Input:
Integrate[1/(x^10*(1 - x^2 + x^4)),x]
Output:
-1/9*1/x^9 - 1/(7*x^7) + 1/(3*x^3) + x^(-1) + ((I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/2])/Sqrt[-6 + (6*I)*Sqrt[3]] + ((-I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/2])/Sqrt[-6 - (6*I)*Sqrt[3]]
Time = 0.60 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {1443, 27, 1604, 27, 1443, 27, 1604, 1447, 1475, 1083, 217, 1478, 25, 1103}
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.
\(\displaystyle \int \frac {1}{x^{10} \left (x^4-x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 1443 |
\(\displaystyle \frac {1}{9} \int \frac {9 \left (1-x^2\right )}{x^8 \left (x^4-x^2+1\right )}dx-\frac {1}{9 x^9}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1-x^2}{x^8 \left (x^4-x^2+1\right )}dx-\frac {1}{9 x^9}\) |
\(\Big \downarrow \) 1604 |
\(\displaystyle -\frac {1}{7} \int \frac {7}{x^4 \left (x^4-x^2+1\right )}dx-\frac {1}{9 x^9}-\frac {1}{7 x^7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {1}{x^4 \left (x^4-x^2+1\right )}dx-\frac {1}{9 x^9}-\frac {1}{7 x^7}\) |
\(\Big \downarrow \) 1443 |
\(\displaystyle -\frac {1}{3} \int \frac {3 \left (1-x^2\right )}{x^2 \left (x^4-x^2+1\right )}dx-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {1-x^2}{x^2 \left (x^4-x^2+1\right )}dx-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}\) |
\(\Big \downarrow \) 1604 |
\(\displaystyle \int \frac {x^2}{x^4-x^2+1}dx-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}\) |
\(\Big \downarrow \) 1447 |
\(\displaystyle -\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx+\frac {1}{2} \int \frac {x^2+1}{x^4-x^2+1}dx-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}\) |
\(\Big \downarrow \) 1475 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {3} x+1}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {3} x+1}dx\right )-\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx+\frac {1}{2} \left (-\int \frac {1}{-\left (2 x-\sqrt {3}\right )^2-1}d\left (2 x-\sqrt {3}\right )-\int \frac {1}{-\left (2 x+\sqrt {3}\right )^2-1}d\left (2 x+\sqrt {3}\right )\right )-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {1}{2} \int \frac {1-x^2}{x^4-x^2+1}dx+\frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}\) |
\(\Big \downarrow \) 1478 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx}{2 \sqrt {3}}+\frac {\int -\frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx}{2 \sqrt {3}}\right )+\frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx}{2 \sqrt {3}}-\frac {\int \frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx}{2 \sqrt {3}}\right )+\frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{x}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right )\right )-\frac {1}{9 x^9}-\frac {1}{7 x^7}+\frac {1}{3 x^3}+\frac {1}{2} \left (\frac {\log \left (x^2-\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{2 \sqrt {3}}\right )+\frac {1}{x}\) |
Input:
Int[1/(x^10*(1 - x^2 + x^4)),x]
Output:
-1/9*1/x^9 - 1/(7*x^7) + 1/(3*x^3) + x^(-1) + (-ArcTan[Sqrt[3] - 2*x] + Ar cTan[Sqrt[3] + 2*x])/2 + (Log[1 - Sqrt[3]*x + x^2]/(2*Sqrt[3]) - Log[1 + S qrt[3]*x + x^2]/(2*Sqrt[3]))/2
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_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
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[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a/c, 2]}, Simp[1/2 Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b ^2 - 4*a*c, 0] && PosQ[a*c]
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[((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*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 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])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {x^{8}+\frac {1}{3} x^{6}-\frac {1}{7} x^{2}-\frac {1}{9}}{x^{9}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (6 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{2}\) | \(50\) |
default | \(-\frac {1}{9 x^{9}}-\frac {1}{7 x^{7}}+\frac {1}{3 x^{3}}+\frac {1}{x}-\frac {\sqrt {3}\, \left (-\frac {\ln \left (x^{2}-\sqrt {3}\, x +1\right )}{2}-\sqrt {3}\, \arctan \left (2 x -\sqrt {3}\right )\right )}{6}-\frac {\sqrt {3}\, \left (\frac {\ln \left (x^{2}+\sqrt {3}\, x +1\right )}{2}-\sqrt {3}\, \arctan \left (2 x +\sqrt {3}\right )\right )}{6}\) | \(87\) |
Input:
int(1/x^10/(x^4-x^2+1),x,method=_RETURNVERBOSE)
Output:
(x^8+1/3*x^6-1/7*x^2-1/9)/x^9+1/2*sum(_R*ln(6*_R^3+_R+x),_R=RootOf(9*_Z^4+ 3*_Z^2+1))
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=-\frac {21 \, \sqrt {3} x^{9} \log \left (x^{2} + \sqrt {3} x + 1\right ) - 21 \, \sqrt {3} x^{9} \log \left (x^{2} - \sqrt {3} x + 1\right ) - 126 \, x^{9} \arctan \left (2 \, x + \sqrt {3}\right ) + 126 \, x^{9} \arctan \left (-2 \, x + \sqrt {3}\right ) - 252 \, x^{8} - 84 \, x^{6} + 36 \, x^{2} + 28}{252 \, x^{9}} \] Input:
integrate(1/x^10/(x^4-x^2+1),x, algorithm="fricas")
Output:
-1/252*(21*sqrt(3)*x^9*log(x^2 + sqrt(3)*x + 1) - 21*sqrt(3)*x^9*log(x^2 - sqrt(3)*x + 1) - 126*x^9*arctan(2*x + sqrt(3)) + 126*x^9*arctan(-2*x + sq rt(3)) - 252*x^8 - 84*x^6 + 36*x^2 + 28)/x^9
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=\frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} + \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{2} + \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{2} + \frac {63 x^{8} + 21 x^{6} - 9 x^{2} - 7}{63 x^{9}} \] Input:
integrate(1/x**10/(x**4-x**2+1),x)
Output:
sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/1 2 + atan(2*x - sqrt(3))/2 + atan(2*x + sqrt(3))/2 + (63*x**8 + 21*x**6 - 9 *x**2 - 7)/(63*x**9)
\[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=\int { \frac {1}{{\left (x^{4} - x^{2} + 1\right )} x^{10}} \,d x } \] Input:
integrate(1/x^10/(x^4-x^2+1),x, algorithm="maxima")
Output:
1/63*(63*x^8 + 21*x^6 - 9*x^2 - 7)/x^9 + integrate(x^2/(x^4 - x^2 + 1), x)
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {63 \, x^{8} + 21 \, x^{6} - 9 \, x^{2} - 7}{63 \, x^{9}} + \frac {1}{2} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{2} \, \arctan \left (2 \, x - \sqrt {3}\right ) \] Input:
integrate(1/x^10/(x^4-x^2+1),x, algorithm="giac")
Output:
1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) - 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) + 1/63*(63*x^8 + 21*x^6 - 9*x^2 - 7)/x^9 + 1/2*arctan(2*x + sqrt(3)) + 1/2*arctan(2*x - sqrt(3))
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=-\frac {-x^8-\frac {x^6}{3}+\frac {x^2}{7}+\frac {1}{9}}{x^9}-\mathrm {atan}\left (\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\mathrm {atan}\left (\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \] Input:
int(1/(x^10*(x^4 - x^2 + 1)),x)
Output:
atan(x/2 + (3^(1/2)*x*1i)/2)*((3^(1/2)*1i)/6 + 1/2) - atan(x/2 - (3^(1/2)* x*1i)/2)*((3^(1/2)*1i)/6 - 1/2) - (x^2/7 - x^6/3 - x^8 + 1/9)/x^9
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^{10} \left (1-x^2+x^4\right )} \, dx=\frac {-126 \mathit {atan} \left (\sqrt {3}-2 x \right ) x^{9}+126 \mathit {atan} \left (\sqrt {3}+2 x \right ) x^{9}+21 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}\, x +x^{2}+1\right ) x^{9}-21 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}\, x +x^{2}+1\right ) x^{9}+252 x^{8}+84 x^{6}-36 x^{2}-28}{252 x^{9}} \] Input:
int(1/x^10/(x^4-x^2+1),x)
Output:
( - 126*atan(sqrt(3) - 2*x)*x**9 + 126*atan(sqrt(3) + 2*x)*x**9 + 21*sqrt( 3)*log( - sqrt(3)*x + x**2 + 1)*x**9 - 21*sqrt(3)*log(sqrt(3)*x + x**2 + 1 )*x**9 + 252*x**8 + 84*x**6 - 36*x**2 - 28)/(252*x**9)