\(\int \frac {(-2+x^6) (4+x^6) \sqrt [4]{-2+2 x^4+x^6}}{x^6 (-4-x^4+2 x^6)} \, dx\) [1496]

Optimal result

Integrand size = 42, antiderivative size = 104 \[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=\frac {\sqrt [4]{-2+2 x^4+x^6} \left (-4+9 x^4+2 x^6\right )}{10 x^5}+\frac {1}{4} \sqrt [4]{\frac {5}{2}} \arctan \left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right )-\frac {1}{4} \sqrt [4]{\frac {5}{2}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right ) \]

[Out]

1/10*(x^6+2*x^4-2)^(1/4)*(2*x^6+9*x^4-4)/x^5+1/8*5^(1/4)*2^(3/4)*arctan(1/2*5^(1/4)*2^(3/4)*x/(x^6+2*x^4-2)^(1
/4))-1/8*5^(1/4)*2^(3/4)*arctanh(1/2*5^(1/4)*2^(3/4)*x/(x^6+2*x^4-2)^(1/4))

Rubi [F]

\[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=\int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx \]

[In]

Int[((-2 + x^6)*(4 + x^6)*(-2 + 2*x^4 + x^6)^(1/4))/(x^6*(-4 - x^4 + 2*x^6)),x]

[Out]

Defer[Int][(-2 + 2*x^4 + x^6)^(1/4), x]/2 + 2*Defer[Int][(-2 + 2*x^4 + x^6)^(1/4)/x^6, x] - Defer[Int][(-2 + 2
*x^4 + x^6)^(1/4)/x^2, x]/2 - Defer[Int][(x^2*(-2 + 2*x^4 + x^6)^(1/4))/(-4 - x^4 + 2*x^6), x]/2 + (3*Defer[In
t][(x^4*(-2 + 2*x^4 + x^6)^(1/4))/(-4 - x^4 + 2*x^6), x])/2

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} \sqrt [4]{-2+2 x^4+x^6}+\frac {2 \sqrt [4]{-2+2 x^4+x^6}}{x^6}-\frac {\sqrt [4]{-2+2 x^4+x^6}}{2 x^2}+\frac {x^2 \left (1-3 x^2\right ) \sqrt [4]{-2+2 x^4+x^6}}{2 \left (4+x^4-2 x^6\right )}\right ) \, dx \\ & = \frac {1}{2} \int \sqrt [4]{-2+2 x^4+x^6} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^2} \, dx+\frac {1}{2} \int \frac {x^2 \left (1-3 x^2\right ) \sqrt [4]{-2+2 x^4+x^6}}{4+x^4-2 x^6} \, dx+2 \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^6} \, dx \\ & = \frac {1}{2} \int \sqrt [4]{-2+2 x^4+x^6} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^2} \, dx+\frac {1}{2} \int \left (-\frac {x^2 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6}+\frac {3 x^4 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6}\right ) \, dx+2 \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^6} \, dx \\ & = \frac {1}{2} \int \sqrt [4]{-2+2 x^4+x^6} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^2} \, dx-\frac {1}{2} \int \frac {x^2 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6} \, dx+\frac {3}{2} \int \frac {x^4 \sqrt [4]{-2+2 x^4+x^6}}{-4-x^4+2 x^6} \, dx+2 \int \frac {\sqrt [4]{-2+2 x^4+x^6}}{x^6} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=\frac {\sqrt [4]{-2+2 x^4+x^6} \left (-4+9 x^4+2 x^6\right )}{10 x^5}+\frac {1}{4} \sqrt [4]{\frac {5}{2}} \arctan \left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right )-\frac {1}{4} \sqrt [4]{\frac {5}{2}} \text {arctanh}\left (\frac {\sqrt [4]{\frac {5}{2}} x}{\sqrt [4]{-2+2 x^4+x^6}}\right ) \]

[In]

Integrate[((-2 + x^6)*(4 + x^6)*(-2 + 2*x^4 + x^6)^(1/4))/(x^6*(-4 - x^4 + 2*x^6)),x]

[Out]

((-2 + 2*x^4 + x^6)^(1/4)*(-4 + 9*x^4 + 2*x^6))/(10*x^5) + ((5/2)^(1/4)*ArcTan[((5/2)^(1/4)*x)/(-2 + 2*x^4 + x
^6)^(1/4)])/4 - ((5/2)^(1/4)*ArcTanh[((5/2)^(1/4)*x)/(-2 + 2*x^4 + x^6)^(1/4)])/4

Maple [A] (verified)

Time = 31.77 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {-5 x^{5} 5^{\frac {1}{4}} \left (\ln \left (\frac {2^{\frac {3}{4}} 5^{\frac {1}{4}} x +2 \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}}}{-2^{\frac {3}{4}} 5^{\frac {1}{4}} x +2 \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} 2^{\frac {1}{4}} 5^{\frac {3}{4}}}{5 x}\right )\right ) 2^{\frac {3}{4}}+8 \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} \left (2 x^{6}+9 x^{4}-4\right )}{80 x^{5}}\)	\(122\)
trager	\(\frac {\left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} \left (2 x^{6}+9 x^{4}-4\right )}{10 x^{5}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2}\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2}\right ) x^{6}-9 \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2}\right ) x^{4}-20 \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2} x^{3}+40 \sqrt {x^{6}+2 x^{4}-2}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2}\right ) x^{2}+80 \left (x^{6}+2 x^{4}-2\right )^{\frac {3}{4}} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2}\right )}{2 x^{6}-x^{4}-4}\right )}{16}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{3} x^{6}-9 \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{3} x^{4}+20 \left (x^{6}+2 x^{4}-2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{2} x^{3}-40 \sqrt {x^{6}+2 x^{4}-2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right ) x^{2}+80 \left (x^{6}+2 x^{4}-2\right )^{\frac {3}{4}} x +4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-40\right )^{3}}{2 x^{6}-x^{4}-4}\right )}{16}\)	\(334\)
risch	\(\text {Expression too large to display}\)	\(1512\)

[In]

int((x^6-2)*(x^6+4)*(x^6+2*x^4-2)^(1/4)/x^6/(2*x^6-x^4-4),x,method=_RETURNVERBOSE)

[Out]

1/80*(-5*x^5*5^(1/4)*(ln((2^(3/4)*5^(1/4)*x+2*(x^6+2*x^4-2)^(1/4))/(-2^(3/4)*5^(1/4)*x+2*(x^6+2*x^4-2)^(1/4)))
+2*arctan(1/5*(x^6+2*x^4-2)^(1/4)/x*2^(1/4)*5^(3/4)))*2^(3/4)+8*(x^6+2*x^4-2)^(1/4)*(2*x^6+9*x^4-4))/x^5

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 83.35 (sec) , antiderivative size = 467, normalized size of antiderivative = 4.49 \[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=-\frac {5 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 10 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} + 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} + 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) - 5 i \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 10 i \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} + 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (-2 i \, x^{6} - 9 i \, x^{4} + 4 i\right )} - 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) + 5 i \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - 10 i \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} + 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 i \, x^{6} + 9 i \, x^{4} - 4 i\right )} - 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) - 5 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} x^{5} \log \left (-\frac {10 \, \sqrt {5} \sqrt {2} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - 10 \cdot 5^{\frac {1}{4}} 2^{\frac {3}{4}} \sqrt {x^{6} + 2 \, x^{4} - 2} x^{2} - 5^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} + 20 \, {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{6} - x^{4} - 4}\right ) - 16 \, {\left (2 \, x^{6} + 9 \, x^{4} - 4\right )} {\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}}}{160 \, x^{5}} \]

[In]

integrate((x^6-2)*(x^6+4)*(x^6+2*x^4-2)^(1/4)/x^6/(2*x^6-x^4-4),x, algorithm="fricas")

[Out]

-1/160*(5*5^(1/4)*2^(3/4)*x^5*log(-(10*sqrt(5)*sqrt(2)*(x^6 + 2*x^4 - 2)^(1/4)*x^3 + 10*5^(1/4)*2^(3/4)*sqrt(x
^6 + 2*x^4 - 2)*x^2 + 5^(3/4)*2^(1/4)*(2*x^6 + 9*x^4 - 4) + 20*(x^6 + 2*x^4 - 2)^(3/4)*x)/(2*x^6 - x^4 - 4)) -
5*I*5^(1/4)*2^(3/4)*x^5*log((10*sqrt(5)*sqrt(2)*(x^6 + 2*x^4 - 2)^(1/4)*x^3 + 10*I*5^(1/4)*2^(3/4)*sqrt(x^6 +
2*x^4 - 2)*x^2 + 5^(3/4)*2^(1/4)*(-2*I*x^6 - 9*I*x^4 + 4*I) - 20*(x^6 + 2*x^4 - 2)^(3/4)*x)/(2*x^6 - x^4 - 4)
) + 5*I*5^(1/4)*2^(3/4)*x^5*log((10*sqrt(5)*sqrt(2)*(x^6 + 2*x^4 - 2)^(1/4)*x^3 - 10*I*5^(1/4)*2^(3/4)*sqrt(x^
6 + 2*x^4 - 2)*x^2 + 5^(3/4)*2^(1/4)*(2*I*x^6 + 9*I*x^4 - 4*I) - 20*(x^6 + 2*x^4 - 2)^(3/4)*x)/(2*x^6 - x^4 -
4)) - 5*5^(1/4)*2^(3/4)*x^5*log(-(10*sqrt(5)*sqrt(2)*(x^6 + 2*x^4 - 2)^(1/4)*x^3 - 10*5^(1/4)*2^(3/4)*sqrt(x^6
+ 2*x^4 - 2)*x^2 - 5^(3/4)*2^(1/4)*(2*x^6 + 9*x^4 - 4) + 20*(x^6 + 2*x^4 - 2)^(3/4)*x)/(2*x^6 - x^4 - 4)) - 1
6*(2*x^6 + 9*x^4 - 4)*(x^6 + 2*x^4 - 2)^(1/4))/x^5

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**6-2)*(x**6+4)*(x**6+2*x**4-2)**(1/4)/x**6/(2*x**6-x**4-4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )} {\left (x^{6} - 2\right )}}{{\left (2 \, x^{6} - x^{4} - 4\right )} x^{6}} \,d x } \]

[In]

integrate((x^6-2)*(x^6+4)*(x^6+2*x^4-2)^(1/4)/x^6/(2*x^6-x^4-4),x, algorithm="maxima")

[Out]

integrate((x^6 + 2*x^4 - 2)^(1/4)*(x^6 + 4)*(x^6 - 2)/((2*x^6 - x^4 - 4)*x^6), x)

Giac [F]

\[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{6} + 4\right )} {\left (x^{6} - 2\right )}}{{\left (2 \, x^{6} - x^{4} - 4\right )} x^{6}} \,d x } \]

[In]

integrate((x^6-2)*(x^6+4)*(x^6+2*x^4-2)^(1/4)/x^6/(2*x^6-x^4-4),x, algorithm="giac")

[Out]

integrate((x^6 + 2*x^4 - 2)^(1/4)*(x^6 + 4)*(x^6 - 2)/((2*x^6 - x^4 - 4)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2+x^6\right ) \left (4+x^6\right ) \sqrt [4]{-2+2 x^4+x^6}}{x^6 \left (-4-x^4+2 x^6\right )} \, dx=\int -\frac {\left (x^6-2\right )\,\left (x^6+4\right )\,{\left (x^6+2\,x^4-2\right )}^{1/4}}{x^6\,\left (-2\,x^6+x^4+4\right )} \,d x \]

[In]

int(-((x^6 - 2)*(x^6 + 4)*(2*x^4 + x^6 - 2)^(1/4))/(x^6*(x^4 - 2*x^6 + 4)),x)

[Out]

int(-((x^6 - 2)*(x^6 + 4)*(2*x^4 + x^6 - 2)^(1/4))/(x^6*(x^4 - 2*x^6 + 4)), x)