Integrand size = 30, antiderivative size = 119 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \sqrt [4]{-2-x^4+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{-x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}} \]
2*(x^6-x^4-2)^(1/4)/x+1/2*arctan(2^(1/2)*x*(x^6-x^4-2)^(1/4)/(-x^2+(x^6-x^ 4-2)^(1/2)))*2^(1/2)-1/2*arctanh(2^(1/2)*x*(x^6-x^4-2)^(1/4)/(x^2+(x^6-x^4 -2)^(1/2)))*2^(1/2)
Time = 0.78 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \sqrt [4]{-2-x^4+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{-x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}} \]
(2*(-2 - x^4 + x^6)^(1/4))/x + ArcTan[(Sqrt[2]*x*(-2 - x^4 + x^6)^(1/4))/( -x^2 + Sqrt[-2 - x^4 + x^6])]/Sqrt[2] - ArcTanh[(Sqrt[2]*x*(-2 - x^4 + x^6 )^(1/4))/(x^2 + Sqrt[-2 - x^4 + x^6])]/Sqrt[2]
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.
| \(\displaystyle \int \frac {\left (x^6+4\right ) \sqrt [4]{x^6-x^4-2}}{x^2 \left (x^6-2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {3 x^4 \sqrt [4]{x^6-x^4-2}}{x^6-2}-\frac {2 \sqrt [4]{x^6-x^4-2}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [6]{2}-x}dx}{2 \sqrt [6]{2}}-\frac {\int \frac {\sqrt [4]{x^6-x^4-2}}{x+\sqrt [6]{2}}dx}{2 \sqrt [6]{2}}-\frac {(-1)^{2/3} \int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [6]{2}-\sqrt [3]{-1} x}dx}{2 \sqrt [6]{2}}-\frac {(-1)^{2/3} \int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [3]{-1} x+\sqrt [6]{2}}dx}{2 \sqrt [6]{2}}+\frac {\sqrt [3]{-1} \int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [6]{2}-(-1)^{2/3} x}dx}{2 \sqrt [6]{2}}+\frac {\sqrt [3]{-1} \int \frac {\sqrt [4]{x^6-x^4-2}}{(-1)^{2/3} x+\sqrt [6]{2}}dx}{2 \sqrt [6]{2}}-2 \int \frac {\sqrt [4]{x^6-x^4-2}}{x^2}dx\) |
3.18.68.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 11.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.33
| method | result | size |
| pseudoelliptic | \(\frac {-\ln \left (\frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}-x^{4}-2}}{\sqrt {x^{6}-x^{4}-2}-\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}}\right ) \sqrt {2}\, x -2 \arctan \left (\frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}\, x -2 \arctan \left (\frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}\, x +8 \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}}}{4 x}\) | \(158\) |
| trager | \(\frac {2 \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{6}-x^{4}-2}\, x^{2}+2 \left (x^{6}-x^{4}-2\right )^{\frac {3}{4}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}-2}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{6}-x^{4}-2}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-2 \left (x^{6}-x^{4}-2\right )^{\frac {3}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}-2}\right )}{2}\) | \(251\) |
| risch | \(\text {Expression too large to display}\) | \(1370\) |
1/4*(-ln(((x^6-x^4-2)^(1/4)*2^(1/2)*x+x^2+(x^6-x^4-2)^(1/2))/((x^6-x^4-2)^ (1/2)-(x^6-x^4-2)^(1/4)*2^(1/2)*x+x^2))*2^(1/2)*x-2*arctan(((x^6-x^4-2)^(1 /4)*2^(1/2)+x)/x)*2^(1/2)*x-2*arctan(((x^6-x^4-2)^(1/4)*2^(1/2)-x)/x)*2^(1 /2)*x+8*(x^6-x^4-2)^(1/4))/x
Result contains complex when optimal does not.
Time = 64.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.99 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {-\left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} + \left (2 i - 2\right ) \, x^{4} + 2 i - 2\right )}}{x^{6} - 2}\right ) + \left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} - \left (2 i - 2\right ) \, x^{4} - 2 i + 2\right )}}{x^{6} - 2}\right ) + \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} - \left (2 i + 2\right ) \, x^{4} - 2 i - 2\right )}}{x^{6} - 2}\right ) - \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} + \left (2 i + 2\right ) \, x^{4} + 2 i + 2\right )}}{x^{6} - 2}\right ) + 16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}}{8 \, x} \]
1/8*(-(I + 1)*sqrt(2)*x*log((4*I*(x^6 - x^4 - 2)^(1/4)*x^3 + (2*I + 2)*sqr t(2)*sqrt(x^6 - x^4 - 2)*x^2 + 4*(x^6 - x^4 - 2)^(3/4)*x + sqrt(2)*(-(I - 1)*x^6 + (2*I - 2)*x^4 + 2*I - 2))/(x^6 - 2)) + (I + 1)*sqrt(2)*x*log((4*I *(x^6 - x^4 - 2)^(1/4)*x^3 - (2*I + 2)*sqrt(2)*sqrt(x^6 - x^4 - 2)*x^2 + 4 *(x^6 - x^4 - 2)^(3/4)*x + sqrt(2)*((I - 1)*x^6 - (2*I - 2)*x^4 - 2*I + 2) )/(x^6 - 2)) + (I - 1)*sqrt(2)*x*log((-4*I*(x^6 - x^4 - 2)^(1/4)*x^3 - (2* I - 2)*sqrt(2)*sqrt(x^6 - x^4 - 2)*x^2 + 4*(x^6 - x^4 - 2)^(3/4)*x + sqrt( 2)*((I + 1)*x^6 - (2*I + 2)*x^4 - 2*I - 2))/(x^6 - 2)) - (I - 1)*sqrt(2)*x *log((-4*I*(x^6 - x^4 - 2)^(1/4)*x^3 + (2*I - 2)*sqrt(2)*sqrt(x^6 - x^4 - 2)*x^2 + 4*(x^6 - x^4 - 2)^(3/4)*x + sqrt(2)*(-(I + 1)*x^6 + (2*I + 2)*x^4 + 2*I + 2))/(x^6 - 2)) + 16*(x^6 - x^4 - 2)^(1/4))/x
\[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int \frac {\left (x^{6} + 4\right ) \sqrt [4]{x^{6} - x^{4} - 2}}{x^{2} \left (x^{6} - 2\right )}\, dx \]
\[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )}}{{\left (x^{6} - 2\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )}}{{\left (x^{6} - 2\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int \frac {\left (x^6+4\right )\,{\left (x^6-x^4-2\right )}^{1/4}}{x^2\,\left (x^6-2\right )} \,d x \]