Integrand size = 40, antiderivative size = 70 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\frac {2 \sqrt [4]{1-x^4+x^6} \left (1+9 x^4+x^6\right )}{5 x^5}+2 \arctan \left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right ) \]
2/5*(x^6-x^4+1)^(1/4)*(x^6+9*x^4+1)/x^5+2*arctan(x/(x^6-x^4+1)^(1/4))-2*ar ctanh(x/(x^6-x^4+1)^(1/4))
Time = 1.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\frac {2 \sqrt [4]{1-x^4+x^6} \left (1+9 x^4+x^6\right )}{5 x^5}+2 \arctan \left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right ) \]
(2*(1 - x^4 + x^6)^(1/4)*(1 + 9*x^4 + x^6))/(5*x^5) + 2*ArcTan[x/(1 - x^4 + x^6)^(1/4)] - 2*ArcTanh[x/(1 - x^4 + x^6)^(1/4)]
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.
\(\displaystyle \int \frac {\left (x^6-2\right ) \left (x^6+1\right ) \sqrt [4]{x^6-x^4+1}}{x^6 \left (x^6-2 x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 2461 |
\(\displaystyle \int \left (\frac {\left (x^6-2\right ) \sqrt [4]{x^6-x^4+1} \left (x^6+1\right )}{x^6 \left (1-x^2\right )}+\frac {\left (x^6-2\right ) \sqrt [4]{x^6-x^4+1} \left (x^6+1\right )}{x^4 \left (x^4-x^2-1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \sqrt [4]{x^6-x^4+1}dx+\int \frac {\sqrt [4]{x^6-x^4+1}}{-x-1}dx+\int \frac {\sqrt [4]{x^6-x^4+1}}{x-1}dx-2 \int \frac {\sqrt [4]{x^6-x^4+1}}{x^6}dx+\frac {2 i \int \frac {\sqrt [4]{x^6-x^4+1}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x}dx}{\sqrt {\sqrt {5}-1}}-\frac {2 \int \frac {\sqrt [4]{x^6-x^4+1}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x}dx}{\sqrt {1+\sqrt {5}}}+\frac {2 i \int \frac {\sqrt [4]{x^6-x^4+1}}{\sqrt {2} x+i \sqrt {-1+\sqrt {5}}}dx}{\sqrt {\sqrt {5}-1}}-\frac {2 \int \frac {\sqrt [4]{x^6-x^4+1}}{\sqrt {2} x+\sqrt {1+\sqrt {5}}}dx}{\sqrt {1+\sqrt {5}}}-4 \int \frac {\sqrt [4]{x^6-x^4+1}}{x^2}dx\) |
3.10.24.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u, (Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[ Qx, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 15.86 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.47
method | result | size |
pseudoelliptic | \(\frac {5 \ln \left (\frac {\left (x^{6}-x^{4}+1\right )^{\frac {1}{4}}-x}{x}\right ) x^{5}-5 \ln \left (\frac {\left (x^{6}-x^{4}+1\right )^{\frac {1}{4}}+x}{x}\right ) x^{5}-10 \arctan \left (\frac {\left (x^{6}-x^{4}+1\right )^{\frac {1}{4}}}{x}\right ) x^{5}+2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} \left (x^{6}+9 x^{4}+1\right )}{5 x^{5}}\) | \(103\) |
trager | \(\frac {2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} \left (x^{6}+9 x^{4}+1\right )}{5 x^{5}}-\ln \left (-\frac {x^{6}+2 \left (x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +2 \sqrt {x^{6}-x^{4}+1}\, x^{2}+2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}+1}{\left (x -1\right ) \left (1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}-x^{4}+1}\, x^{2}-2 \left (x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )\) | \(216\) |
risch | \(\text {Expression too large to display}\) | \(1238\) |
1/5*(5*ln(((x^6-x^4+1)^(1/4)-x)/x)*x^5-5*ln(((x^6-x^4+1)^(1/4)+x)/x)*x^5-1 0*arctan((x^6-x^4+1)^(1/4)/x)*x^5+2*(x^6-x^4+1)^(1/4)*(x^6+9*x^4+1))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (62) = 124\).
Time = 111.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.20 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\frac {5 \, x^{5} \arctan \left (\frac {2 \, {\left ({\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{6} - 2 \, x^{4} + 1}\right ) + 5 \, x^{5} \log \left (\frac {x^{6} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{6} - x^{4} + 1} x^{2} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} - 2 \, x^{4} + 1}\right ) + 2 \, {\left (x^{6} + 9 \, x^{4} + 1\right )} {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]
1/5*(5*x^5*arctan(2*((x^6 - x^4 + 1)^(1/4)*x^3 + (x^6 - x^4 + 1)^(3/4)*x)/ (x^6 - 2*x^4 + 1)) + 5*x^5*log((x^6 - 2*(x^6 - x^4 + 1)^(1/4)*x^3 + 2*sqrt (x^6 - x^4 + 1)*x^2 - 2*(x^6 - x^4 + 1)^(3/4)*x + 1)/(x^6 - 2*x^4 + 1)) + 2*(x^6 + 9*x^4 + 1)*(x^6 - x^4 + 1)^(1/4))/x^5
Timed out. \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{6} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} - 2 \, x^{4} + 1\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{6} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} - 2 \, x^{4} + 1\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\int \frac {\left (x^6+1\right )\,\left (x^6-2\right )\,{\left (x^6-x^4+1\right )}^{1/4}}{x^6\,\left (x^6-2\,x^4+1\right )} \,d x \]