Integrand size = 19, antiderivative size = 354 \[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx=\frac {2^{2/3} \arctan \left (\frac {a-2 \sqrt [3]{\frac {2}{-1+\sqrt {5}}} x}{\sqrt {3} a}\right )}{\sqrt {15} \left (-1+\sqrt {5}\right )^{2/3} a^5}-\frac {2^{2/3} \arctan \left (\frac {a+2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} x}{\sqrt {3} a}\right )}{\sqrt {15} \left (1+\sqrt {5}\right )^{2/3} a^5}+\frac {2^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right )^{2/3} a^5}-\frac {2^{2/3} \log \left (\sqrt [3]{-1+\sqrt {5}} a+\sqrt [3]{2} x\right )}{3 \sqrt {5} \left (-1+\sqrt {5}\right )^{2/3} a^5}+\frac {\log \left (\left (-1+\sqrt {5}\right )^{2/3} a^2-\sqrt [3]{2 \left (-1+\sqrt {5}\right )} a x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {5} \left (-1+\sqrt {5}\right )^{2/3} a^5}-\frac {\log \left (\left (1+\sqrt {5}\right )^{2/3} a^2+\sqrt [3]{2 \left (1+\sqrt {5}\right )} a x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {5} \left (1+\sqrt {5}\right )^{2/3} a^5} \] Output:
1/15*2^(2/3)*arctan(1/3*(a-2*2^(1/3)*(1/(5^(1/2)-1))^(1/3)*x)*3^(1/2)/a)*1 5^(1/2)/(5^(1/2)-1)^(2/3)/a^5-1/15*2^(2/3)*arctan(1/3*(a+2*2^(1/3)*(1/(5^( 1/2)+1))^(1/3)*x)*3^(1/2)/a)*15^(1/2)/(5^(1/2)+1)^(2/3)/a^5+1/15*2^(2/3)*l n((5^(1/2)+1)^(1/3)*a-2^(1/3)*x)*5^(1/2)/(5^(1/2)+1)^(2/3)/a^5-1/15*2^(2/3 )*ln((5^(1/2)-1)^(1/3)*a+2^(1/3)*x)*5^(1/2)/(5^(1/2)-1)^(2/3)/a^5+1/30*ln( (5^(1/2)-1)^(2/3)*a^2-(-2+2*5^(1/2))^(1/3)*a*x+2^(2/3)*x^2)*2^(2/3)*5^(1/2 )/(5^(1/2)-1)^(2/3)/a^5-1/30*ln((5^(1/2)+1)^(2/3)*a^2+(2+2*5^(1/2))^(1/3)* a*x+2^(2/3)*x^2)*2^(2/3)*5^(1/2)/(5^(1/2)+1)^(2/3)/a^5
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.14 \[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx=-\frac {1}{3} \text {RootSum}\left [a^6+a^3 \text {$\#$1}^3-\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{a^3 \text {$\#$1}^2-2 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(-a^6 - a^3*x^3 + x^6)^(-1),x]
Output:
-1/3*RootSum[a^6 + a^3*#1^3 - #1^6 & , Log[x - #1]/(a^3*#1^2 - 2*#1^5) & ]
Time = 0.75 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1685, 750, 16, 25, 1142, 25, 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.
| \(\displaystyle \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx\) |
| \(\Big \downarrow \) 1685 |
| \(\displaystyle \frac {\int \frac {1}{x^3-\frac {1}{2} \left (1+\sqrt {5}\right ) a^3}dx}{\sqrt {5} a^3}-\frac {\int \frac {1}{x^3-\frac {1}{2} \left (1-\sqrt {5}\right ) a^3}dx}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \int -\frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} a+x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx}{3 a^2}+\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \int \frac {1}{x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} a}dx}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \int \frac {2^{2/3} \sqrt [3]{-1+\sqrt {5}} a-x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \int \frac {1}{\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} a+x}dx}{3 a^2}}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \int -\frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} a+x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx}{3 a^2}+\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \int \frac {2^{2/3} \sqrt [3]{-1+\sqrt {5}} a-x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \log \left (\sqrt [3]{\sqrt {5}-1} a+\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 a^2}-\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \int \frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} a+x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \int \frac {2^{2/3} \sqrt [3]{-1+\sqrt {5}} a-x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \log \left (\sqrt [3]{\sqrt {5}-1} a+\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 a^2}-\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \left (\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} a \int \frac {1}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx+\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} a+2 x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx\right )}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \left (\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (\sqrt {5}-1\right )} a \int \frac {1}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx-\frac {1}{2} \int -\frac {\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} a-2 x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx\right )}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \log \left (\sqrt [3]{\sqrt {5}-1} a+\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 a^2}-\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \left (\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} a \int \frac {1}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx+\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} a+2 x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx\right )}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \left (\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (\sqrt {5}-1\right )} a \int \frac {1}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx+\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} a-2 x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx\right )}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \log \left (\sqrt [3]{\sqrt {5}-1} a+\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 a^2}-\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} a+2 x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx-3 \int \frac {1}{-\left (\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} x}{a}+1\right )^2-3}d\left (\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} x}{a}+1\right )\right )}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} a-2 x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx+3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{\frac {2}{-1+\sqrt {5}}} x}{a}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{\frac {2}{-1+\sqrt {5}}} x}{a}\right )\right )}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \log \left (\sqrt [3]{\sqrt {5}-1} a+\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 a^2}-\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} a+2 x}{\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3} a^2+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x a+x^2}dx+\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} x}{a}+1}{\sqrt {3}}\right )\right )}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \left (\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} a-2 x}{\left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right )^{2/3} a^2-\sqrt [3]{\frac {1}{2} \left (-1+\sqrt {5}\right )} x a+x^2}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\frac {2}{\sqrt {5}-1}} x}{a}}{\sqrt {3}}\right )\right )}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \log \left (\sqrt [3]{\sqrt {5}-1} a+\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \log \left (\sqrt [3]{1+\sqrt {5}} a-\sqrt [3]{2} x\right )}{3 a^2}-\frac {\left (\frac {2}{1+\sqrt {5}}\right )^{2/3} \left (\frac {1}{2} \log \left (\left (1+\sqrt {5}\right )^{2/3} a^2+\sqrt [3]{2 \left (1+\sqrt {5}\right )} a x+2^{2/3} x^2\right )+\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {5}}} x}{a}+1}{\sqrt {3}}\right )\right )}{3 a^2}}{\sqrt {5} a^3}-\frac {\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \left (-\frac {1}{2} \log \left (\left (\sqrt {5}-1\right )^{2/3} a^2-\sqrt [3]{2 \left (\sqrt {5}-1\right )} a x+2^{2/3} x^2\right )-\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{\frac {2}{\sqrt {5}-1}} x}{a}}{\sqrt {3}}\right )\right )}{3 a^2}+\frac {\left (\frac {2}{\sqrt {5}-1}\right )^{2/3} \log \left (\sqrt [3]{\sqrt {5}-1} a+\sqrt [3]{2} x\right )}{3 a^2}}{\sqrt {5} a^3}\) |
Input:
Int[(-a^6 - a^3*x^3 + x^6)^(-1),x]
Output:
-((((2/(-1 + Sqrt[5]))^(2/3)*Log[(-1 + Sqrt[5])^(1/3)*a + 2^(1/3)*x])/(3*a ^2) + ((2/(-1 + Sqrt[5]))^(2/3)*(-(Sqrt[3]*ArcTan[(1 - (2*(2/(-1 + Sqrt[5] ))^(1/3)*x)/a)/Sqrt[3]]) - Log[(-1 + Sqrt[5])^(2/3)*a^2 - (2*(-1 + Sqrt[5] ))^(1/3)*a*x + 2^(2/3)*x^2]/2))/(3*a^2))/(Sqrt[5]*a^3)) + (((2/(1 + Sqrt[5 ]))^(2/3)*Log[(1 + Sqrt[5])^(1/3)*a - 2^(1/3)*x])/(3*a^2) - ((2/(1 + Sqrt[ 5]))^(2/3)*(Sqrt[3]*ArcTan[(1 + (2*(2/(1 + Sqrt[5]))^(1/3)*x)/a)/Sqrt[3]] + Log[(1 + Sqrt[5])^(2/3)*a^2 + (2*(1 + Sqrt[5]))^(1/3)*a*x + 2^(2/3)*x^2] /2))/(3*a^2))/(Sqrt[5]*a^3)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^n), x], x] - Simp[ c/q Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.13
| method | result | size |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-a^{3} \textit {\_Z}^{3}-a^{6}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-2 \textit {\_R}^{5}+\textit {\_R}^{2} a^{3}}\right )}{3}\) | \(46\) |
| risch | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-a^{3} \textit {\_Z}^{3}-a^{6}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-2 \textit {\_R}^{5}+\textit {\_R}^{2} a^{3}}\right )}{3}\) | \(46\) |
Input:
int(1/(-a^6-a^3*x^3+x^6),x,method=_RETURNVERBOSE)
Output:
-1/3*sum(1/(-2*_R^5+_R^2*a^3)*ln(x-_R),_R=RootOf(_Z^6-_Z^3*a^3-a^6))
Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (255) = 510\).
Time = 0.08 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.52 \[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx =\text {Too large to display} \] Input:
integrate(1/(-a^6-a^3*x^3+x^6),x, algorithm="fricas")
Output:
-1/6*(1/10)^(1/3)*(sqrt(-3) + 1)*(-(3*sqrt(1/5)*a^15*sqrt(a^(-30)) + 1)/a^ 15)^(1/3)*log(5*(1/10)^(1/3)*(sqrt(-3)*a^6 + a^6 - sqrt(1/5)*(sqrt(-3)*a^2 1 + a^21)*sqrt(a^(-30)))*(-(3*sqrt(1/5)*a^15*sqrt(a^(-30)) + 1)/a^15)^(1/3 ) + 4*x) + 1/6*(1/10)^(1/3)*(sqrt(-3) - 1)*(-(3*sqrt(1/5)*a^15*sqrt(a^(-30 )) + 1)/a^15)^(1/3)*log(-5*(1/10)^(1/3)*(sqrt(-3)*a^6 - a^6 - sqrt(1/5)*(s qrt(-3)*a^21 - a^21)*sqrt(a^(-30)))*(-(3*sqrt(1/5)*a^15*sqrt(a^(-30)) + 1) /a^15)^(1/3) + 4*x) - 1/6*(1/10)^(1/3)*(sqrt(-3) + 1)*((3*sqrt(1/5)*a^15*s qrt(a^(-30)) - 1)/a^15)^(1/3)*log(5*(1/10)^(1/3)*(sqrt(-3)*a^6 + a^6 + sqr t(1/5)*(sqrt(-3)*a^21 + a^21)*sqrt(a^(-30)))*((3*sqrt(1/5)*a^15*sqrt(a^(-3 0)) - 1)/a^15)^(1/3) + 4*x) + 1/6*(1/10)^(1/3)*(sqrt(-3) - 1)*((3*sqrt(1/5 )*a^15*sqrt(a^(-30)) - 1)/a^15)^(1/3)*log(-5*(1/10)^(1/3)*(sqrt(-3)*a^6 - a^6 + sqrt(1/5)*(sqrt(-3)*a^21 - a^21)*sqrt(a^(-30)))*((3*sqrt(1/5)*a^15*s qrt(a^(-30)) - 1)/a^15)^(1/3) + 4*x) + 1/3*(1/10)^(1/3)*(-(3*sqrt(1/5)*a^1 5*sqrt(a^(-30)) + 1)/a^15)^(1/3)*log(5*(1/10)^(1/3)*(sqrt(1/5)*a^21*sqrt(a ^(-30)) - a^6)*(-(3*sqrt(1/5)*a^15*sqrt(a^(-30)) + 1)/a^15)^(1/3) + 2*x) + 1/3*(1/10)^(1/3)*((3*sqrt(1/5)*a^15*sqrt(a^(-30)) - 1)/a^15)^(1/3)*log(-5 *(1/10)^(1/3)*(sqrt(1/5)*a^21*sqrt(a^(-30)) + a^6)*((3*sqrt(1/5)*a^15*sqrt (a^(-30)) - 1)/a^15)^(1/3) + 2*x)
Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.09 \[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx=\frac {\operatorname {RootSum} {\left (91125 t^{6} + 675 t^{3} - 1, \left ( t \mapsto t \log {\left (- 675 t^{4} a - 10 t a + x \right )} \right )\right )}}{a^{5}} \] Input:
integrate(1/(-a**6-a**3*x**3+x**6),x)
Output:
RootSum(91125*_t**6 + 675*_t**3 - 1, Lambda(_t, _t*log(-675*_t**4*a - 10*_ t*a + x)))/a**5
\[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx=\int { -\frac {1}{a^{6} + a^{3} x^{3} - x^{6}} \,d x } \] Input:
integrate(1/(-a^6-a^3*x^3+x^6),x, algorithm="maxima")
Output:
-integrate(1/(a^6 + a^3*x^3 - x^6), x)
\[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx=\int { -\frac {1}{a^{6} + a^{3} x^{3} - x^{6}} \,d x } \] Input:
integrate(1/(-a^6-a^3*x^3+x^6),x, algorithm="giac")
Output:
integrate(-1/(a^6 + a^3*x^3 - x^6), x)
Time = 13.16 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.90 \[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx =\text {Too large to display} \] Input:
int(-1/(a^6 - x^6 + a^3*x^3),x)
Output:
log(6*x + (9*2^(2/3)*5^(1/3)*a^6*(3^(1/2)*1i - 1)*(3*5^(1/2) - 5)^(1/3)*(1 /a^15)^(1/3)*(5^(1/2) + (2^(1/3)*5^(2/3)*a^9*x*(3*5^(1/2) - 5)^(2/3)*(1/a^ 15)^(2/3))/6 + (2^(1/3)*3^(1/2)*5^(2/3)*a^9*x*(3*5^(1/2) - 5)^(2/3)*(1/a^1 5)^(2/3)*1i)/6 - 5))/40)*((3^(1/2)*1i)/2 - 1/2)*((3*5^(1/2) - 5)/(1350*a^1 5))^(1/3) - log(6*x - (9*2^(2/3)*5^(1/3)*a^6*(3^(1/2)*1i + 1)*(3*5^(1/2) - 5)^(1/3)*(1/a^15)^(1/3)*(5^(1/2) + (2^(1/3)*5^(2/3)*a^9*x*(3*5^(1/2) - 5) ^(2/3)*(1/a^15)^(2/3))/6 - (2^(1/3)*3^(1/2)*5^(2/3)*a^9*x*(3*5^(1/2) - 5)^ (2/3)*(1/a^15)^(2/3)*1i)/6 - 5))/40)*((3^(1/2)*1i)/2 + 1/2)*((3*5^(1/2) - 5)/(1350*a^15))^(1/3) + log(6*x - (9*2^(2/3)*5^(1/3)*a^6*(3^(1/2)*1i - 1)* (3*5^(1/2) + 5)^(1/3)*(-1/a^15)^(1/3)*(5^(1/2) - (2^(1/3)*5^(2/3)*a^9*x*(3 *5^(1/2) + 5)^(2/3)*(-1/a^15)^(2/3))/6 - (2^(1/3)*3^(1/2)*5^(2/3)*a^9*x*(3 *5^(1/2) + 5)^(2/3)*(-1/a^15)^(2/3)*1i)/6 + 5))/40)*((3^(1/2)*1i)/2 - 1/2) *(-(3*5^(1/2) + 5)/(1350*a^15))^(1/3) - log(6*x + (9*2^(2/3)*5^(1/3)*a^6*( 3^(1/2)*1i + 1)*(3*5^(1/2) + 5)^(1/3)*(-1/a^15)^(1/3)*(5^(1/2) - (2^(1/3)* 5^(2/3)*a^9*x*(3*5^(1/2) + 5)^(2/3)*(-1/a^15)^(2/3))/6 + (2^(1/3)*3^(1/2)* 5^(2/3)*a^9*x*(3*5^(1/2) + 5)^(2/3)*(-1/a^15)^(2/3)*1i)/6 + 5))/40)*((3^(1 /2)*1i)/2 + 1/2)*(-(3*5^(1/2) + 5)/(1350*a^15))^(1/3) + (1350^(2/3)*log(40 50*x - 1350*5^(1/2)*x - 450*2^(2/3)*5^(1/3)*a^21*(3*5^(1/2) - 5)^(1/3)*(1/ a^15)^(4/3) + 45*2^(2/3)*5^(1/3)*a^21*(3*5^(1/2) - 5)^(4/3)*(1/a^15)^(4/3) )*(3*5^(1/2) - 5)^(1/3)*(1/a^15)^(1/3))/1350 + (1350^(2/3)*log(4050*x +...
\[ \int \frac {1}{-a^6-a^3 x^3+x^6} \, dx=-\left (\int \frac {1}{a^{6}+a^{3} x^{3}-x^{6}}d x \right ) \] Input:
int(1/(-a^6-a^3*x^3+x^6),x)
Output:
- int(1/(a**6 + a**3*x**3 - x**6),x)