Integrand size = 38, antiderivative size = 43 \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\frac {1}{3} \arctan \left (\frac {1-x^3}{\sqrt [4]{1+x^6}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {-1+x^3}{\sqrt [4]{1+x^6}}\right ) \]
Time = 8.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\frac {1}{3} \arctan \left (\frac {1-x^3}{\sqrt [4]{1+x^6}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {-1+x^3}{\sqrt [4]{1+x^6}}\right ) \]
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.
\(\displaystyle \int \frac {x^6+x^3+2}{x \sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )} \, dx\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle \frac {1}{3} \int -\frac {x^6+x^3+2}{x^3 \sqrt [4]{x^6+1} \left (-x^9+4 x^6-5 x^3+4\right )}dx^3\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {x^6+x^3+2}{x^3 \sqrt [4]{x^6+1} \left (-x^9+4 x^6-5 x^3+4\right )}dx^3\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {-x^6+2 x^3-7}{2 \sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}+\frac {1}{2 x^3 \sqrt [4]{x^6+1}}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {7}{2} \int \frac {1}{\sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}dx^3-\int \frac {x^3}{\sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}dx^3+\frac {1}{2} \int \frac {x^6}{\sqrt [4]{x^6+1} \left (x^9-4 x^6+5 x^3-4\right )}dx^3-\frac {1}{2} \arctan \left (\sqrt [4]{x^6+1}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt [4]{x^6+1}\right )\right )\) |
3.6.61.3.1 Defintions of rubi rules used
Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 249.52 (sec) , antiderivative size = 349, normalized size of antiderivative = 8.12
method | result | size |
trager | \(\frac {\ln \left (-\frac {-x^{12}+2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}+4 x^{9}-2 \sqrt {x^{6}+1}\, x^{6}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}-7 x^{6}+4 x^{3} \sqrt {x^{6}+1}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}-2 \left (x^{6}+1\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{6}+1}-2 \left (x^{6}+1\right )^{\frac {1}{4}}-2}{x^{3} \left (x^{9}-4 x^{6}+5 x^{3}-4\right )}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{12}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{9}-2 \left (x^{6}+1\right )^{\frac {1}{4}} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+1}\, x^{6}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+1}\, x^{3}+2 \left (x^{6}+1\right )^{\frac {3}{4}} x^{3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-6 \left (x^{6}+1\right )^{\frac {1}{4}} x^{3}+2 \sqrt {x^{6}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{6}+1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \left (x^{6}+1\right )^{\frac {1}{4}}}{x^{3} \left (x^{9}-4 x^{6}+5 x^{3}-4\right )}\right )}{6}\) | \(349\) |
1/6*ln(-(-x^12+2*(x^6+1)^(1/4)*x^9+4*x^9-2*(x^6+1)^(1/2)*x^6-6*(x^6+1)^(1/ 4)*x^6+2*(x^6+1)^(3/4)*x^3-7*x^6+4*x^3*(x^6+1)^(1/2)+6*(x^6+1)^(1/4)*x^3-2 *(x^6+1)^(3/4)+4*x^3-2*(x^6+1)^(1/2)-2*(x^6+1)^(1/4)-2)/x^3/(x^9-4*x^6+5*x ^3-4))+1/6*RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*x^12+4*RootOf(_Z^2+1)*x^9-2 *(x^6+1)^(1/4)*x^9+2*RootOf(_Z^2+1)*(x^6+1)^(1/2)*x^6-7*RootOf(_Z^2+1)*x^6 +6*(x^6+1)^(1/4)*x^6-4*RootOf(_Z^2+1)*(x^6+1)^(1/2)*x^3+2*(x^6+1)^(3/4)*x^ 3+4*RootOf(_Z^2+1)*x^3-6*(x^6+1)^(1/4)*x^3+2*(x^6+1)^(1/2)*RootOf(_Z^2+1)- 2*(x^6+1)^(3/4)-2*RootOf(_Z^2+1)+2*(x^6+1)^(1/4))/x^3/(x^9-4*x^6+5*x^3-4))
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (34) = 68\).
Time = 6.85 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.88 \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\frac {1}{6} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}\right )}}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\right ) + \frac {1}{6} \, \log \left (-\frac {x^{12} - 4 \, x^{9} + 7 \, x^{6} - 4 \, x^{3} - 2 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )} + 2 \, {\left (x^{6} - 2 \, x^{3} + 1\right )} \sqrt {x^{6} + 1} - 2 \, {\left (x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} + 2}{x^{12} - 4 \, x^{9} + 5 \, x^{6} - 4 \, x^{3}}\right ) \]
1/6*arctan(2*((x^6 + 1)^(3/4)*(x^3 - 1) + (x^9 - 3*x^6 + 3*x^3 - 1)*(x^6 + 1)^(1/4))/(x^12 - 4*x^9 + 5*x^6 - 4*x^3)) + 1/6*log(-(x^12 - 4*x^9 + 7*x^ 6 - 4*x^3 - 2*(x^6 + 1)^(3/4)*(x^3 - 1) + 2*(x^6 - 2*x^3 + 1)*sqrt(x^6 + 1 ) - 2*(x^9 - 3*x^6 + 3*x^3 - 1)*(x^6 + 1)^(1/4) + 2)/(x^12 - 4*x^9 + 5*x^6 - 4*x^3))
Timed out. \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\text {Timed out} \]
\[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\int { \frac {x^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x} \,d x } \]
\[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\int { \frac {x^{6} + x^{3} + 2}{{\left (x^{9} - 4 \, x^{6} + 5 \, x^{3} - 4\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}} x} \,d x } \]
Timed out. \[ \int \frac {2+x^3+x^6}{x \sqrt [4]{1+x^6} \left (-4+5 x^3-4 x^6+x^9\right )} \, dx=\int \frac {x^6+x^3+2}{x\,{\left (x^6+1\right )}^{1/4}\,\left (x^9-4\,x^6+5\,x^3-4\right )} \,d x \]