\(\int \frac {(4+x^6) \sqrt [4]{-2-x^4+x^6}}{x^2 (-2+x^6)} \, dx\) [1768]

Optimal result

Integrand size = 30, antiderivative size = 119 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \sqrt [4]{-2-x^4+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{-x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}} \]

[Out]

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)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {2 \sqrt [4]{-2-x^4+x^6}}{x}+\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{-x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-2-x^4+x^6}}{x^2+\sqrt {-2-x^4+x^6}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

(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]

Maple [A] (verified)

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	\(\text {Expression too large to display}\)	\(1370\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 64.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.99 \[ \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx=\frac {-\left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} + \left (2 i - 2\right ) \, x^{4} + 2 i - 2\right )}}{x^{6} - 2}\right ) + \left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} - \left (2 i - 2\right ) \, x^{4} - 2 i + 2\right )}}{x^{6} - 2}\right ) + \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} - \left (2 i + 2\right ) \, x^{4} - 2 i - 2\right )}}{x^{6} - 2}\right ) - \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-4 i \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} - x^{4} - 2} x^{2} + 4 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} + \left (2 i + 2\right ) \, x^{4} + 2 i + 2\right )}}{x^{6} - 2}\right ) + 16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}}{8 \, x} \]

[In]

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

[Out]

1/8*(-(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)) - (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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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