Integrand size = 30, antiderivative size = 119 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \sqrt [4]{-2-x^4+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{-x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}} \] Output:
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 = 0.76 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \sqrt [4]{-2-x^4+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{-x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}} \] Input:
Integrate[((4 + x^6)*(-2 - x^4 + x^6)^(1/4))/(x^2*(-2 + x^6)),x]
Output:
(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 |
| \(\displaystyle -\frac {\int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [6]{2}-x}dx}{2 \sqrt [6]{2}}-\frac {\int \frac {\sqrt [4]{x^6-x^4-2}}{x+\sqrt [6]{2}}dx}{2 \sqrt [6]{2}}-\frac {(-1)^{2/3} \int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [6]{2}-\sqrt [3]{-1} x}dx}{2 \sqrt [6]{2}}-\frac {(-1)^{2/3} \int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [3]{-1} x+\sqrt [6]{2}}dx}{2 \sqrt [6]{2}}+\frac {\sqrt [3]{-1} \int \frac {\sqrt [4]{x^6-x^4-2}}{\sqrt [6]{2}-(-1)^{2/3} x}dx}{2 \sqrt [6]{2}}+\frac {\sqrt [3]{-1} \int \frac {\sqrt [4]{x^6-x^4-2}}{(-1)^{2/3} x+\sqrt [6]{2}}dx}{2 \sqrt [6]{2}}-2 \int \frac {\sqrt [4]{x^6-x^4-2}}{x^2}dx\) |
Input:
Int[((4 + x^6)*(-2 - x^4 + x^6)^(1/4))/(x^2*(-2 + x^6)),x]
Output:
$Aborted
Time = 14.80 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.33
| method | result | size |
| pseudoelliptic | \(\frac {-\ln \left (\frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}-x^{4}-2}}{-\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{6}-x^{4}-2}}\right ) \sqrt {2}\, x -2 \arctan \left (\frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}\, x -2 \arctan \left (\frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}\, x +8 \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}}}{4 x}\) | \(158\) |
| trager | \(\frac {2 \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{6}-x^{4}-2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{6}-x^{4}-2\right )^{\frac {3}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}-2}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-2 \sqrt {x^{6}-x^{4}-2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-2 \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{6}-x^{4}-2\right )^{\frac {3}{4}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{6}-2}\right )}{2}\) | \(250\) |
| risch | \(\text {Expression too large to display}\) | \(1372\) |
Input:
int((x^6+4)*(x^6-x^4-2)^(1/4)/x^2/(x^6-2),x,method=_RETURNVERBOSE)
Output:
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) ^(1/4)*2^(1/2)*x+x^2+(x^6-x^4-2)^(1/2)))*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
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (103) = 206\).
Time = 61.44 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.61 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \, \sqrt {2} x \arctan \left (\frac {x^{12} - 4 \, x^{6} + 2 \, \sqrt {2} {\left (x^{7} - 4 \, x^{5} - 2 \, x\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} - 4 \, x^{7} - 6 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} + 4}{x^{12} - 16 \, x^{10} + 16 \, x^{8} - 4 \, x^{6} + 32 \, x^{4} + 4}\right ) + 2 \, \sqrt {2} x \arctan \left (-\frac {x^{12} - 4 \, x^{6} - 2 \, \sqrt {2} {\left (x^{7} - 4 \, x^{5} - 2 \, x\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} - 4 \, x^{7} - 6 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} + 4}{x^{12} - 16 \, x^{10} + 16 \, x^{8} - 4 \, x^{6} + 32 \, x^{4} + 4}\right ) - \sqrt {2} x \log \left (\frac {x^{6} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2}{x^{6} - 2}\right ) + \sqrt {2} x \log \left (\frac {x^{6} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2}{x^{6} - 2}\right ) + 16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}}{8 \, x} \] Input:
integrate((x^6+4)*(x^6-x^4-2)^(1/4)/x^2/(x^6-2),x, algorithm="fricas")
Output:
1/8*(2*sqrt(2)*x*arctan((x^12 - 4*x^6 + 2*sqrt(2)*(x^7 - 4*x^5 - 2*x)*(x^6 - x^4 - 2)^(3/4) + 2*sqrt(2)*(3*x^9 - 4*x^7 - 6*x^3)*(x^6 - x^4 - 2)^(1/4 ) + 4*(x^8 - 2*x^2)*sqrt(x^6 - x^4 - 2) + 4)/(x^12 - 16*x^10 + 16*x^8 - 4* x^6 + 32*x^4 + 4)) + 2*sqrt(2)*x*arctan(-(x^12 - 4*x^6 - 2*sqrt(2)*(x^7 - 4*x^5 - 2*x)*(x^6 - x^4 - 2)^(3/4) - 2*sqrt(2)*(3*x^9 - 4*x^7 - 6*x^3)*(x^ 6 - x^4 - 2)^(1/4) + 4*(x^8 - 2*x^2)*sqrt(x^6 - x^4 - 2) + 4)/(x^12 - 16*x ^10 + 16*x^8 - 4*x^6 + 32*x^4 + 4)) - sqrt(2)*x*log((x^6 + 2*sqrt(2)*(x^6 - x^4 - 2)^(1/4)*x^3 + 4*sqrt(x^6 - x^4 - 2)*x^2 + 2*sqrt(2)*(x^6 - x^4 - 2)^(3/4)*x - 2)/(x^6 - 2)) + sqrt(2)*x*log((x^6 - 2*sqrt(2)*(x^6 - x^4 - 2 )^(1/4)*x^3 + 4*sqrt(x^6 - x^4 - 2)*x^2 - 2*sqrt(2)*(x^6 - x^4 - 2)^(3/4)* x - 2)/(x^6 - 2)) + 16*(x^6 - x^4 - 2)^(1/4))/x
\[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int \frac {\left (x^{6} + 4\right ) \sqrt [4]{x^{6} - x^{4} - 2}}{x^{2} \left (x^{6} - 2\right )}\, dx \] Input:
integrate((x**6+4)*(x**6-x**4-2)**(1/4)/x**2/(x**6-2),x)
Output:
Integral((x**6 + 4)*(x**6 - x**4 - 2)**(1/4)/(x**2*(x**6 - 2)), x)
\[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )}}{{\left (x^{6} - 2\right )} x^{2}} \,d x } \] Input:
integrate((x^6+4)*(x^6-x^4-2)^(1/4)/x^2/(x^6-2),x, algorithm="maxima")
Output:
integrate((x^6 - x^4 - 2)^(1/4)*(x^6 + 4)/((x^6 - 2)*x^2), x)
\[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )}}{{\left (x^{6} - 2\right )} x^{2}} \,d x } \] Input:
integrate((x^6+4)*(x^6-x^4-2)^(1/4)/x^2/(x^6-2),x, algorithm="giac")
Output:
integrate((x^6 - x^4 - 2)^(1/4)*(x^6 + 4)/((x^6 - 2)*x^2), x)
Timed out. \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\int \frac {\left (x^6+4\right )\,{\left (x^6-x^4-2\right )}^{1/4}}{x^2\,\left (x^6-2\right )} \,d x \] Input:
int(((x^6 + 4)*(x^6 - x^4 - 2)^(1/4))/(x^2*(x^6 - 2)),x)
Output:
int(((x^6 + 4)*(x^6 - x^4 - 2)^(1/4))/(x^2*(x^6 - 2)), x)
\[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \left (x^{6}-x^{4}-2\right )^{\frac {1}{4}}-\left (\int \frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} x^{8}}{x^{12}-x^{10}-4 x^{6}+2 x^{4}+4}d x \right ) x -4 \left (\int \frac {\left (x^{6}-x^{4}-2\right )^{\frac {1}{4}} x^{2}}{x^{12}-x^{10}-4 x^{6}+2 x^{4}+4}d x \right ) x}{x} \] Input:
int((x^6+4)*(x^6-x^4-2)^(1/4)/x^2/(x^6-2),x)
Output:
(2*(x**6 - x**4 - 2)**(1/4) - int(((x**6 - x**4 - 2)**(1/4)*x**8)/(x**12 - x**10 - 4*x**6 + 2*x**4 + 4),x)*x - 4*int(((x**6 - x**4 - 2)**(1/4)*x**2) /(x**12 - x**10 - 4*x**6 + 2*x**4 + 4),x)*x)/x