Integrand size = 16, antiderivative size = 418 \[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=-\frac {1}{2 x^2}-\frac {\left (i-\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18 \sqrt [3]{2} \left (1+i \sqrt {3}\right )^{2/3}} \] Output:
-1/2/x^2-1/6*(I-3^(1/2))*arctan(1/3*(1+2*x/(1/2-1/2*I*3^(1/2))^(1/3))*3^(1 /2))*2^(2/3)/(1-I*3^(1/2))^(2/3)+1/6*(3^(1/2)+I)*arctan(1/3*(1+2*x/(1/2+1/ 2*I*3^(1/2))^(1/3))*3^(1/2))*2^(2/3)/(1+I*3^(1/2))^(2/3)-1/18*(3-I*3^(1/2) )*ln((1-I*3^(1/2))^(1/3)-2^(1/3)*x)*2^(2/3)/(1-I*3^(1/2))^(2/3)-1/18*(3+I* 3^(1/2))*ln((1+I*3^(1/2))^(1/3)-2^(1/3)*x)*2^(2/3)/(1+I*3^(1/2))^(2/3)+1/3 6*(3-I*3^(1/2))*ln((1-I*3^(1/2))^(2/3)+(2-2*I*3^(1/2))^(1/3)*x+2^(2/3)*x^2 )*2^(2/3)/(1-I*3^(1/2))^(2/3)+1/36*(3+I*3^(1/2))*ln((1+I*3^(1/2))^(2/3)+(2 +2*I*3^(1/2))^(1/3)*x+2^(2/3)*x^2)*2^(2/3)/(1+I*3^(1/2))^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=-\frac {1}{2 x^2}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[1/(x^3*(1 - x^3 + x^6)),x]
Output:
-1/2*1/x^2 - RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3) /(-#1^2 + 2*#1^5) & ]/3
Time = 0.54 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.85, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1704, 27, 1752, 750, 16, 25, 1142, 1082, 217, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (x^6-x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 1704 |
| \(\displaystyle \frac {1}{2} \int \frac {2 \left (1-x^3\right )}{x^6-x^3+1}dx-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1-x^3}{x^6-x^3+1}dx-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle -\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {1}{x^3+\frac {1}{2} \left (-1-i \sqrt {3}\right )}dx-\frac {1}{6} \left (3-i \sqrt {3}\right ) \int \frac {1}{x^3+\frac {1}{2} \left (-1+i \sqrt {3}\right )}dx-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1-i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}+\frac {\int \frac {1}{x-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1+i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}+\frac {\int \frac {1}{x-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1-i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}+\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\int -\frac {x+2^{2/3} \sqrt [3]{1+i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}+\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\int \frac {x+2^{2/3} \sqrt [3]{1-i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\int \frac {x+2^{2/3} \sqrt [3]{1+i \sqrt {3}}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} \int \frac {1}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} \int \frac {1}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx-3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx-3 \int \frac {1}{-\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )^2-3}d\left (\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+1\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}dx+\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\frac {1}{2} \int \frac {2 x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}{x^2+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}dx+\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{2 x^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {1}{6} \left (3-i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )+\frac {1}{2} \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{6} \left (3+i \sqrt {3}\right ) \left (\frac {\log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )+\frac {1}{2} \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}}\right )-\frac {1}{2 x^2}\) |
Input:
Int[1/(x^3*(1 - x^3 + x^6)),x]
Output:
-1/2*1/x^2 - ((3 - I*Sqrt[3])*(Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x]/(3*( (1 - I*Sqrt[3])/2)^(2/3)) - (Sqrt[3]*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2) ^(1/3))/Sqrt[3]] + Log[(1 - I*Sqrt[3])^(2/3) + (2*(1 - I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2]/2)/(3*((1 - I*Sqrt[3])/2)^(2/3))))/6 - ((3 + I*Sqrt[3])*(L og[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x]/(3*((1 + I*Sqrt[3])/2)^(2/3)) - (Sqr t[3]*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]] + Log[(1 + I*Sq rt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2]/2)/(3*((1 + I*Sq rt[3])/2)^(2/3))))/6
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_ Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1) )), x] - Simp[1/(a*d^n*(m + 1)) Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{ a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.09
| method | result | size |
| risch | \(-\frac {1}{2 x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (27 \textit {\_Z}^{6}+9 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (9 \textit {\_R}^{4}+3 \textit {\_R} +x \right )\right )}{3}\) | \(38\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}-\frac {1}{2 x^{2}}\) | \(50\) |
Input:
int(1/x^3/(x^6-x^3+1),x,method=_RETURNVERBOSE)
Output:
-1/2/x^2+1/3*sum(_R*ln(9*_R^4+3*_R+x),_R=RootOf(27*_Z^6+9*_Z^3+1))
Time = 0.07 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=\frac {2 \, x^{2} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (3 \, {\left (\sqrt {-\frac {1}{3}} + 1\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} + 2 \, x\right ) + 2 \, x^{2} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (-3 \, {\left (\sqrt {-\frac {1}{3}} - 1\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} + 2 \, x\right ) - {\left (\sqrt {-3} x^{2} + x^{2}\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (-3 \, {\left (\sqrt {-\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} + \sqrt {-3} + 1\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} + 4 \, x\right ) + {\left (\sqrt {-3} x^{2} - x^{2}\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (3 \, {\left (\sqrt {-\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} + \sqrt {-3} - 1\right )} {\left (\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} + 4 \, x\right ) - {\left (\sqrt {-3} x^{2} + x^{2}\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (3 \, {\left (\sqrt {-\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} - \sqrt {-3} - 1\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} + 4 \, x\right ) + {\left (\sqrt {-3} x^{2} - x^{2}\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} \log \left (-3 \, {\left (\sqrt {-\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} - \sqrt {-3} + 1\right )} {\left (-\frac {1}{6} \, \sqrt {-\frac {1}{3}} - \frac {1}{6}\right )}^{\frac {1}{3}} + 4 \, x\right ) - 3}{6 \, x^{2}} \] Input:
integrate(1/x^3/(x^6-x^3+1),x, algorithm="fricas")
Output:
1/6*(2*x^2*(1/6*sqrt(-1/3) - 1/6)^(1/3)*log(3*(sqrt(-1/3) + 1)*(1/6*sqrt(- 1/3) - 1/6)^(1/3) + 2*x) + 2*x^2*(-1/6*sqrt(-1/3) - 1/6)^(1/3)*log(-3*(sqr t(-1/3) - 1)*(-1/6*sqrt(-1/3) - 1/6)^(1/3) + 2*x) - (sqrt(-3)*x^2 + x^2)*( 1/6*sqrt(-1/3) - 1/6)^(1/3)*log(-3*(sqrt(-1/3)*(sqrt(-3) + 1) + sqrt(-3) + 1)*(1/6*sqrt(-1/3) - 1/6)^(1/3) + 4*x) + (sqrt(-3)*x^2 - x^2)*(1/6*sqrt(- 1/3) - 1/6)^(1/3)*log(3*(sqrt(-1/3)*(sqrt(-3) - 1) + sqrt(-3) - 1)*(1/6*sq rt(-1/3) - 1/6)^(1/3) + 4*x) - (sqrt(-3)*x^2 + x^2)*(-1/6*sqrt(-1/3) - 1/6 )^(1/3)*log(3*(sqrt(-1/3)*(sqrt(-3) + 1) - sqrt(-3) - 1)*(-1/6*sqrt(-1/3) - 1/6)^(1/3) + 4*x) + (sqrt(-3)*x^2 - x^2)*(-1/6*sqrt(-1/3) - 1/6)^(1/3)*l og(-3*(sqrt(-1/3)*(sqrt(-3) - 1) - sqrt(-3) + 1)*(-1/6*sqrt(-1/3) - 1/6)^( 1/3) + 4*x) - 3)/x^2
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.07 \[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=\operatorname {RootSum} {\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log {\left (729 t^{4} + 9 t + x \right )} \right )\right )} - \frac {1}{2 x^{2}} \] Input:
integrate(1/x**3/(x**6-x**3+1),x)
Output:
RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(729*_t**4 + 9*_t + x))) - 1/(2*x**2)
\[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} - x^{3} + 1\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(x^6-x^3+1),x, algorithm="maxima")
Output:
-1/2/x^2 - integrate((x^3 - 1)/(x^6 - x^3 + 1), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (260) = 520\).
Time = 0.15 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x^3/(x^6-x^3+1),x, algorithm="giac")
Output:
1/9*(sqrt(3)*cos(4/9*pi)^4 - 6*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^2 + sqrt( 3)*sin(4/9*pi)^4 + 4*cos(4/9*pi)^3*sin(4/9*pi) - 4*cos(4/9*pi)*sin(4/9*pi) ^3 + 2*sqrt(3)*cos(4/9*pi) + 2*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*c os(4/9*pi) + 2*x)/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) + 1/9*(sqrt(3)*cos( 2/9*pi)^4 - 6*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^2 + sqrt(3)*sin(2/9*pi)^4 + 4*cos(2/9*pi)^3*sin(2/9*pi) - 4*cos(2/9*pi)*sin(2/9*pi)^3 + 2*sqrt(3)*co s(2/9*pi) + 2*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi) + 2*x) /((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) + 1/9*(sqrt(3)*cos(1/9*pi)^4 - 6*sqr t(3)*cos(1/9*pi)^2*sin(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^4 - 4*cos(1/9*pi)^3 *sin(1/9*pi) + 4*cos(1/9*pi)*sin(1/9*pi)^3 - 2*sqrt(3)*cos(1/9*pi) + 2*sin (1/9*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9*pi) - 2*x)/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/18*(4*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi) - 4*sqrt (3)*cos(4/9*pi)*sin(4/9*pi)^3 - cos(4/9*pi)^4 + 6*cos(4/9*pi)^2*sin(4/9*pi )^2 - sin(4/9*pi)^4 + 2*sqrt(3)*sin(4/9*pi) - 2*cos(4/9*pi))*log((-I*sqrt( 3)*cos(4/9*pi) - cos(4/9*pi))*x + x^2 + 1) + 1/18*(4*sqrt(3)*cos(2/9*pi)^3 *sin(2/9*pi) - 4*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^3 - cos(2/9*pi)^4 + 6*cos (2/9*pi)^2*sin(2/9*pi)^2 - sin(2/9*pi)^4 + 2*sqrt(3)*sin(2/9*pi) - 2*cos(2 /9*pi))*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*x + x^2 + 1) - 1/18*(4* sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi) - 4*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^3 + cos(1/9*pi)^4 - 6*cos(1/9*pi)^2*sin(1/9*pi)^2 + sin(1/9*pi)^4 - 2*sqrt(...
Time = 19.76 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=\frac {\ln \left (x-\frac {\left (-\frac {27}{2}+\frac {\sqrt {3}\,9{}\mathrm {i}}{2}\right )\,{\left (-36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{54}\right )\,{\left (-36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x+\frac {\left (\frac {27}{2}+\frac {\sqrt {3}\,9{}\mathrm {i}}{2}\right )\,{\left (-36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{54}\right )\,{\left (-36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {1}{2\,x^2}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )\,\left (\frac {3\,\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}^3}{16}+27\right )}{108}\right )\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )\,\left (\frac {3\,\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}^3}{16}-27\right )}{108}\right )\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{2/3}\,3^{5/6}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,3^{5/6}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{6}\right )\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \] Input:
int(1/(x^3*(x^6 - x^3 + 1)),x)
Output:
(log(x - (((3^(1/2)*9i)/2 - 27/2)*(- 3^(1/2)*12i - 36)^(1/3))/54)*(- 3^(1/ 2)*12i - 36)^(1/3))/18 + (log(x + (((3^(1/2)*9i)/2 + 27/2)*(3^(1/2)*12i - 36)^(1/3))/54)*(3^(1/2)*12i - 36)^(1/3))/18 - 1/(2*x^2) - (2^(2/3)*log(x - (2^(2/3)*(- 3^(1/2)*1i - 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i)*((3*(3^(1/2)*1i + 3)*(3^(1/3) + 3^(5/6)*1i)^3)/16 + 27))/108)*(- 3^(1/2)*1i - 3)^(1/3)*(3^ (1/3) + 3^(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(2/3)*(3^(1/2)*1i - 3)^(1/3) *(3^(1/3) - 3^(5/6)*1i)*((3*(3^(1/2)*1i - 3)*(3^(1/3) - 3^(5/6)*1i)^3)/16 - 27))/108)*(3^(1/2)*1i - 3)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36 - (2^(2/3)*l og(x + (2^(2/3)*3^(5/6)*(- 3^(1/2)*1i - 3)^(1/3)*1i)/6)*(- 3^(1/2)*1i - 3) ^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36 - (2^(2/3)*log(x - (2^(2/3)*3^(5/6)*(3^( 1/2)*1i - 3)^(1/3)*1i)/6)*(3^(1/2)*1i - 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i))/3 6
\[ \int \frac {1}{x^3 \left (1-x^3+x^6\right )} \, dx=\frac {-2 \left (\int \frac {x^{3}}{x^{6}-x^{3}+1}d x \right ) x^{2}+2 \left (\int \frac {1}{x^{6}-x^{3}+1}d x \right ) x^{2}-1}{2 x^{2}} \] Input:
int(1/x^3/(x^6-x^3+1),x)
Output:
( - 2*int(x**3/(x**6 - x**3 + 1),x)*x**2 + 2*int(1/(x**6 - x**3 + 1),x)*x* *2 - 1)/(2*x**2)