\(\int \frac {(-2+x^6) (1+x^6) \sqrt [4]{1-x^4+x^6}}{x^6 (1-2 x^4+x^6)} \, dx\) [924]

Optimal result

Integrand size = 40, antiderivative size = 70 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\frac {2 \sqrt [4]{1-x^4+x^6} \left (1+9 x^4+x^6\right )}{5 x^5}+2 \arctan \left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

Defer[Int][(1 - x^4 + x^6)^(1/4), x] + Defer[Int][(1 - x^4 + x^6)^(1/4)/(-1 - x), x] + Defer[Int][(1 - x^4 + x
^6)^(1/4)/(-1 + x), x] - 2*Defer[Int][(1 - x^4 + x^6)^(1/4)/x^6, x] - 4*Defer[Int][(1 - x^4 + x^6)^(1/4)/x^2,
x] + ((2*I)*Defer[Int][(1 - x^4 + x^6)^(1/4)/(I*Sqrt[-1 + Sqrt[5]] - Sqrt[2]*x), x])/Sqrt[-1 + Sqrt[5]] - (2*D
efer[Int][(1 - x^4 + x^6)^(1/4)/(Sqrt[1 + Sqrt[5]] - Sqrt[2]*x), x])/Sqrt[1 + Sqrt[5]] + ((2*I)*Defer[Int][(1
- x^4 + x^6)^(1/4)/(I*Sqrt[-1 + Sqrt[5]] + Sqrt[2]*x), x])/Sqrt[-1 + Sqrt[5]] - (2*Defer[Int][(1 - x^4 + x^6)^
(1/4)/(Sqrt[1 + Sqrt[5]] + Sqrt[2]*x), x])/Sqrt[1 + Sqrt[5]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt [4]{1-x^4+x^6}+\frac {\sqrt [4]{1-x^4+x^6}}{-1-x}+\frac {\sqrt [4]{1-x^4+x^6}}{-1+x}-\frac {2 \sqrt [4]{1-x^4+x^6}}{x^6}-\frac {4 \sqrt [4]{1-x^4+x^6}}{x^2}+\frac {2 \left (-1+2 x^2\right ) \sqrt [4]{1-x^4+x^6}}{-1-x^2+x^4}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )+2 \int \frac {\left (-1+2 x^2\right ) \sqrt [4]{1-x^4+x^6}}{-1-x^2+x^4} \, dx-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )+2 \int \left (\frac {2 \sqrt [4]{1-x^4+x^6}}{-1-\sqrt {5}+2 x^2}+\frac {2 \sqrt [4]{1-x^4+x^6}}{-1+\sqrt {5}+2 x^2}\right ) \, dx-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+4 \int \frac {\sqrt [4]{1-x^4+x^6}}{-1-\sqrt {5}+2 x^2} \, dx+4 \int \frac {\sqrt [4]{1-x^4+x^6}}{-1+\sqrt {5}+2 x^2} \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+4 \int \left (\frac {i \sqrt [4]{1-x^4+x^6}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {i \sqrt [4]{1-x^4+x^6}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+4 \int \left (\frac {\sqrt {1+\sqrt {5}} \sqrt [4]{1-x^4+x^6}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {\sqrt {1+\sqrt {5}} \sqrt [4]{1-x^4+x^6}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^6} \, dx\right )-4 \int \frac {\sqrt [4]{1-x^4+x^6}}{x^2} \, dx+\frac {(2 i) \int \frac {\sqrt [4]{1-x^4+x^6}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {-1+\sqrt {5}}}+\frac {(2 i) \int \frac {\sqrt [4]{1-x^4+x^6}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {-1+\sqrt {5}}}-\frac {2 \int \frac {\sqrt [4]{1-x^4+x^6}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {1+\sqrt {5}}}-\frac {2 \int \frac {\sqrt [4]{1-x^4+x^6}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {1+\sqrt {5}}}+\int \sqrt [4]{1-x^4+x^6} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-x^4+x^6}}{-1+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\frac {2 \sqrt [4]{1-x^4+x^6} \left (1+9 x^4+x^6\right )}{5 x^5}+2 \arctan \left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^4+x^6}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 15.86 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.47

method	result	size
pseudoelliptic	\(\frac {5 \ln \left (\frac {\left (x^{6}-x^{4}+1\right )^{\frac {1}{4}}-x}{x}\right ) x^{5}-5 \ln \left (\frac {\left (x^{6}-x^{4}+1\right )^{\frac {1}{4}}+x}{x}\right ) x^{5}-10 \arctan \left (\frac {\left (x^{6}-x^{4}+1\right )^{\frac {1}{4}}}{x}\right ) x^{5}+2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} \left (x^{6}+9 x^{4}+1\right )}{5 x^{5}}\)	\(103\)
trager	\(\frac {2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} \left (x^{6}+9 x^{4}+1\right )}{5 x^{5}}-\ln \left (-\frac {x^{6}+2 \left (x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +2 \sqrt {x^{6}-x^{4}+1}\, x^{2}+2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}+1}{\left (x -1\right ) \left (1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}-x^{4}+1}\, x^{2}-2 \left (x^{6}-x^{4}+1\right )^{\frac {3}{4}} x +2 \left (x^{6}-x^{4}+1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{4}-x^{2}-1\right )}\right )\)	\(216\)
risch	\(\text {Expression too large to display}\)	\(1238\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (62) = 124\).

Time = 111.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.20 \[ \int \frac {\left (-2+x^6\right ) \left (1+x^6\right ) \sqrt [4]{1-x^4+x^6}}{x^6 \left (1-2 x^4+x^6\right )} \, dx=\frac {5 \, x^{5} \arctan \left (\frac {2 \, {\left ({\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{6} - 2 \, x^{4} + 1}\right ) + 5 \, x^{5} \log \left (\frac {x^{6} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{6} - x^{4} + 1} x^{2} - 2 \, {\left (x^{6} - x^{4} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} - 2 \, x^{4} + 1}\right ) + 2 \, {\left (x^{6} + 9 \, x^{4} + 1\right )} {\left (x^{6} - x^{4} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]

[In]

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

[Out]

1/5*(5*x^5*arctan(2*((x^6 - x^4 + 1)^(1/4)*x^3 + (x^6 - x^4 + 1)^(3/4)*x)/(x^6 - 2*x^4 + 1)) + 5*x^5*log((x^6
- 2*(x^6 - x^4 + 1)^(1/4)*x^3 + 2*sqrt(x^6 - x^4 + 1)*x^2 - 2*(x^6 - x^4 + 1)^(3/4)*x + 1)/(x^6 - 2*x^4 + 1))
+ 2*(x^6 + 9*x^4 + 1)*(x^6 - x^4 + 1)^(1/4))/x^5

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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