\(\int \frac {(-2+x^2) (-1+x^2) \sqrt [4]{-1+x^2+x^4}}{x^6 (-1+x^2+2 x^4)} \, dx\) [1814]

Optimal result

Integrand size = 38, antiderivative size = 123 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(4*(-1 + x^2 + x^4)^(1/4)*AppellF1[-1/2, -1/4, -1/4, 1/2, (-2*x^2)/(1 - Sqrt[5]), (-2*x^2)/(1 + Sqrt[5])])/(x*
(1 + (2*x^2)/(1 - Sqrt[5]))^(1/4)*(1 + (2*x^2)/(1 + Sqrt[5]))^(1/4)) - ((1 + (2*x^2)/(1 + Sqrt[5]))^(5/4)*(-1
+ x^2 + x^4)^(1/4)*Hypergeometric2F1[-3/2, -1/4, -1/2, (-2*(x^2/(1 - Sqrt[5]) - x^2/(1 + Sqrt[5])))/(1 + (2*x^
2)/(1 + Sqrt[5]))])/(3*x^3*(1 + (2*x^2)/(1 - Sqrt[5]))^(1/4)) - (4*(1 + (2*x^2)/(1 + Sqrt[5]))*(-1 + x^2 + x^4
)^(1/4)*((3*(1 + Sqrt[5]) - (13 + 3*Sqrt[5])*x^2 + 2*(1 + Sqrt[5])*x^4)*Gamma[-1/4]*Hypergeometric2F1[-1/4, 1,
-1/2, (-2*Sqrt[5]*x^2)/(2 - (1 + Sqrt[5])*x^2)] - 4*x^2*(5 + Sqrt[5] + 2*Sqrt[5]*x^2)*Gamma[3/4]*Hypergeometr
ic2F1[3/4, 2, 1/2, (-2*Sqrt[5]*x^2)/(2 - (1 + Sqrt[5])*x^2)]))/(15*(3 + Sqrt[5])*x^5*(1 - Sqrt[5] + 2*x^2)*Gam
ma[-1/4]) + 2*Defer[Int][(-1 + x^2 + x^4)^(1/4)/(1 + x^2), x] + 4*Defer[Int][(-1 + x^2 + x^4)^(1/4)/(-1 + 2*x^
2), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \sqrt [4]{-1+x^2+x^4}}{x^6}+\frac {\sqrt [4]{-1+x^2+x^4}}{x^4}-\frac {4 \sqrt [4]{-1+x^2+x^4}}{x^2}+\frac {2 \sqrt [4]{-1+x^2+x^4}}{1+x^2}+\frac {4 \sqrt [4]{-1+x^2+x^4}}{-1+2 x^2}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{x^6} \, dx\right )+2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx-4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx+\int \frac {\sqrt [4]{-1+x^2+x^4}}{x^4} \, dx \\ & = 2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx+\frac {\sqrt [4]{-1+x^2+x^4} \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^4} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (2 \sqrt [4]{-1+x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^6} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (4 \sqrt [4]{-1+x^2+x^4}\right ) \int \frac {\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}{x^2} \, dx}{\sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}} \\ & = \frac {4 \sqrt [4]{-1+x^2+x^4} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{4},-\frac {1}{4},\frac {1}{2},-\frac {2 x^2}{1-\sqrt {5}},-\frac {2 x^2}{1+\sqrt {5}}\right )}{x \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^2}{1+\sqrt {5}}}}-\frac {\left (1+\frac {2 x^2}{1+\sqrt {5}}\right )^{5/4} \sqrt [4]{-1+x^2+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},-\frac {1}{2},-\frac {2 \left (\frac {x^2}{1-\sqrt {5}}-\frac {x^2}{1+\sqrt {5}}\right )}{1+\frac {2 x^2}{1+\sqrt {5}}}\right )}{3 x^3 \sqrt [4]{1+\frac {2 x^2}{1-\sqrt {5}}}}-\frac {4 \left (1+\frac {2 x^2}{1+\sqrt {5}}\right ) \sqrt [4]{-1+x^2+x^4} \left (\left (3 \left (1+\sqrt {5}\right )-\left (13+3 \sqrt {5}\right ) x^2+2 \left (1+\sqrt {5}\right ) x^4\right ) \operatorname {Gamma}\left (-\frac {1}{4}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,-\frac {1}{2},-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )-4 x^2 \left (5+\sqrt {5}+2 \sqrt {5} x^2\right ) \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},2,\frac {1}{2},-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )\right )}{15 \left (3+\sqrt {5}\right ) x^5 \left (1-\sqrt {5}+2 x^2\right ) \operatorname {Gamma}\left (-\frac {1}{4}\right )}+2 \int \frac {\sqrt [4]{-1+x^2+x^4}}{1+x^2} \, dx+4 \int \frac {\sqrt [4]{-1+x^2+x^4}}{-1+2 x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 5.48 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(\frac {-5 x^{5} \left (2 \arctan \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )+2 \arctan \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+\ln \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}+x^{2}-1}}{\sqrt {x^{4}+x^{2}-1}-\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}}\right )\right ) \sqrt {2}+4 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{10 x^{5}}\)	\(151\)
trager	\(\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{5 x^{5}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )\)	\(239\)
risch	\(\frac {\frac {18}{5} x^{8}+\frac {16}{5} x^{6}-\frac {18}{5} x^{4}+\frac {4}{5} x^{2}-\frac {2}{5}}{x^{5} \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}+\frac {\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{2}-1\right ) \left (x^{2}+1\right ) \left (x^{4}+x^{2}-1\right )^{2}}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{8}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{2}-1\right ) \left (x^{2}+1\right ) \left (x^{4}+x^{2}-1\right )^{2}}\right )\right ) {\left (\left (x^{4}+x^{2}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}\)	\(849\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.86 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.78 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {-\left (5 i + 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x - i \, x^{2} + i}{2 \, x^{4} + x^{2} - 1}\right ) + \left (5 i - 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + i \, x^{2} - i}{2 \, x^{4} + x^{2} - 1}\right ) - \left (5 i - 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + i \, x^{2} - i}{2 \, x^{4} + x^{2} - 1}\right ) + \left (5 i + 5\right ) \, \sqrt {2} x^{5} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x - i \, x^{2} + i}{2 \, x^{4} + x^{2} - 1}\right ) + 8 \, {\left (9 \, x^{4} - x^{2} + 1\right )} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}}}{20 \, x^{5}} \]

[In]

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

[Out]

1/20*(-(5*I + 5)*sqrt(2)*x^5*log(((I + 1)*sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 2*sqrt(x^4 + x^2 - 1)*x^2 - (I -
1)*sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x - I*x^2 + I)/(2*x^4 + x^2 - 1)) + (5*I - 5)*sqrt(2)*x^5*log((-(I - 1)*sqrt
(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 2*sqrt(x^4 + x^2 - 1)*x^2 + (I + 1)*sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x + I*x^2 -
I)/(2*x^4 + x^2 - 1)) - (5*I - 5)*sqrt(2)*x^5*log(((I - 1)*sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 2*sqrt(x^4 + x^
2 - 1)*x^2 - (I + 1)*sqrt(2)*(x^4 + x^2 - 1)^(3/4)*x + I*x^2 - I)/(2*x^4 + x^2 - 1)) + (5*I + 5)*sqrt(2)*x^5*l
og((-(I + 1)*sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 2*sqrt(x^4 + x^2 - 1)*x^2 + (I - 1)*sqrt(2)*(x^4 + x^2 - 1)^(
3/4)*x - I*x^2 + I)/(2*x^4 + x^2 - 1)) + 8*(9*x^4 - x^2 + 1)*(x^4 + x^2 - 1)^(1/4))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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