\(\int \frac {(1+x^2) (1-3 x^2+x^4)}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} (1-x-3 x^2+x^3+x^4)} \, dx\) [2348]

Optimal result

Integrand size = 58, antiderivative size = 186 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {\left (-1+x+x^2\right ) \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{2 x}-\frac {3 \text {arctanh}\left (\frac {\sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {1}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {1}{2 \sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right )+\frac {1}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5+\sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right ) \]

[Out]

1/2*(x^2+x-1)*((2*x^2+x-2)/(x^2+x-1))^(1/2)/x-3/4*arctanh(1/2*((2*x^2+x-2)/(x^2+x-1))^(1/2)*2^(1/2))*2^(1/2)+1
/5*(10-2*5^(1/2))^(1/2)*arctanh(1/10*(50+10*5^(1/2))^(1/2)*((2*x^2+x-2)/(x^2+x-1))^(1/2))+1/5*(10+2*5^(1/2))^(
1/2)*arctanh(2^(1/2)/(5+5^(1/2))^(1/2)*((2*x^2+x-2)/(x^2+x-1))^(1/2))

Rubi [F]

\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 4.57 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.41 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {-20 \sqrt {-1+x+x^2}+10 x \sqrt {-1+x+x^2}+20 x^2 \sqrt {-1+x+x^2}-15 \sqrt {2} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {-2+x+2 x^2}}{\sqrt {2} \sqrt {-1+x+x^2}}\right )+4 \sqrt {2 \left (5-\sqrt {5}\right )} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \sqrt {-2+x+2 x^2}}{\sqrt {-1+x+x^2}}\right )+4 \sqrt {2 \left (5+\sqrt {5}\right )} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {-\left (\left (-5+\sqrt {5}\right ) \left (-2+x+2 x^2\right )\right )}}{\sqrt {10} \sqrt {-1+x+x^2}}\right )}{20 x \sqrt {-1+x+x^2} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}} \]

[In]

Integrate[((1 + x^2)*(1 - 3*x^2 + x^4))/(x^2*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)]*(1 - x - 3*x^2 + x^3 + x^4)
),x]

[Out]

(-20*Sqrt[-1 + x + x^2] + 10*x*Sqrt[-1 + x + x^2] + 20*x^2*Sqrt[-1 + x + x^2] - 15*Sqrt[2]*x*Sqrt[-2 + x + 2*x
^2]*ArcTanh[Sqrt[-2 + x + 2*x^2]/(Sqrt[2]*Sqrt[-1 + x + x^2])] + 4*Sqrt[2*(5 - Sqrt[5])]*x*Sqrt[-2 + x + 2*x^2
]*ArcTanh[(Sqrt[(5 + Sqrt[5])/10]*Sqrt[-2 + x + 2*x^2])/Sqrt[-1 + x + x^2]] + 4*Sqrt[2*(5 + Sqrt[5])]*x*Sqrt[-
2 + x + 2*x^2]*ArcTanh[Sqrt[-((-5 + Sqrt[5])*(-2 + x + 2*x^2))]/(Sqrt[10]*Sqrt[-1 + x + x^2])])/(20*x*Sqrt[-1
+ x + x^2]*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)])

Maple [A] (verified)

Time = 6.06 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.42

method	result	size
risch	\(\frac {2 x^{2}+x -2}{2 x \sqrt {\frac {2 x^{2}+x -2}{x^{2}+x -1}}}+\frac {\left (-\frac {3 \sqrt {2}\, \ln \left (\frac {4 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}+\sqrt {2}\, \left (4 x^{2}+3 x -4\right )}{x}\right )}{8}+\frac {\operatorname {arctanh}\left (\frac {\left (4 x^{2}+3 x -4\right ) \sqrt {5}+2 x^{2}+x -2}{2 \sqrt {10+2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) \sqrt {10-2 \sqrt {5}}}{10}+\frac {\operatorname {arctanh}\left (\frac {\left (4 x^{2}+3 x -4\right ) \sqrt {5}-2 x^{2}-x +2}{2 \sqrt {10-2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) \sqrt {10+2 \sqrt {5}}}{10}\right ) \sqrt {\left (2 x^{2}+x -2\right ) \left (x^{2}+x -1\right )}}{\sqrt {\frac {2 x^{2}+x -2}{x^{2}+x -1}}\, \left (x^{2}+x -1\right )}\)	\(265\)
default	\(-\frac {\left (2 x^{2}+x -2\right ) \left (15 \sqrt {2}\, \ln \left (\frac {4 \sqrt {2}\, x^{2}+3 x \sqrt {2}-4 \sqrt {2}+4 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}{x}\right ) x -4 \sqrt {10-2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {5}\, x^{2}+3 x \sqrt {5}+2 x^{2}-4 \sqrt {5}+x -2}{2 \sqrt {10+2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) x -4 \sqrt {10+2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {5}\, x^{2}+3 x \sqrt {5}-2 x^{2}-4 \sqrt {5}-x +2}{2 \sqrt {10-2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) x -20 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}\right )}{40 \sqrt {\frac {2 x^{2}+x -2}{x^{2}+x -1}}\, \sqrt {\left (2 x^{2}+x -2\right ) \left (x^{2}+x -1\right )}\, x}\)	\(278\)
trager	\(\text {Expression too large to display}\)	\(1371\)

[In]

int((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 723 vs. \(2 (145) = 290\).

Time = 0.50 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.89 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {4 \, x \sqrt {2 \, \sqrt {5} + 10} \log \left (\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} - \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {2 \, \sqrt {5} + 10} + 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} - \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) - 4 \, x \sqrt {2 \, \sqrt {5} + 10} \log \left (-\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} - \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {2 \, \sqrt {5} + 10} - 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} - \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) + 4 \, x \sqrt {-2 \, \sqrt {5} + 10} \log \left (\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} + \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {-2 \, \sqrt {5} + 10} + 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} + \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) - 4 \, x \sqrt {-2 \, \sqrt {5} + 10} \log \left (-\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} + \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {-2 \, \sqrt {5} + 10} - 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} + \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) + 15 \, \sqrt {2} x \log \left (-\frac {32 \, x^{4} + 48 \, x^{3} - 47 \, x^{2} - 4 \, \sqrt {2} {\left (4 \, x^{4} + 7 \, x^{3} - 5 \, x^{2} - 7 \, x + 4\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}} - 48 \, x + 32}{x^{2}}\right ) + 40 \, {\left (x^{2} + x - 1\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{80 \, x} \]

[In]

integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x, algorithm="fricas")

[Out]

1/80*(4*x*sqrt(2*sqrt(5) + 10)*log(((20575*x^4 + 50235*x^3 - 15795*x^2 - sqrt(5)*(10237*x^4 + 22677*x^3 - 9661
*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(2*sqrt(5) + 10) + 20*(1627*x^4 + 4593*x^3 - 288*x^2 - sqrt(5)*
(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3
- 3*x^2 - x + 1)) - 4*x*sqrt(2*sqrt(5) + 10)*log(-((20575*x^4 + 50235*x^3 - 15795*x^2 - sqrt(5)*(10237*x^4 +
22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(2*sqrt(5) + 10) - 20*(1627*x^4 + 4593*x^3 - 28
8*x^2 - sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x -
1)))/(x^4 + x^3 - 3*x^2 - x + 1)) + 4*x*sqrt(-2*sqrt(5) + 10)*log(((20575*x^4 + 50235*x^3 - 15795*x^2 + sqrt(
5)*(10237*x^4 + 22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(-2*sqrt(5) + 10) + 20*(1627*x^
4 + 4593*x^3 - 288*x^2 + sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 +
x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) - 4*x*sqrt(-2*sqrt(5) + 10)*log(-((20575*x^4 + 50235*x^3 -
15795*x^2 + sqrt(5)*(10237*x^4 + 22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(-2*sqrt(5) +
10) - 20*(1627*x^4 + 4593*x^3 - 288*x^2 + sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 16
27)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) + 15*sqrt(2)*x*log(-(32*x^4 + 48*x^3 - 4
7*x^2 - 4*sqrt(2)*(4*x^4 + 7*x^3 - 5*x^2 - 7*x + 4)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)) - 48*x + 32)/x^2) + 40
*(x^2 + x - 1)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/x

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**2+1)*(x**4-3*x**2+1)/x**2/((2*x**2+x-2)/(x**2+x-1))**(1/2)/(x**4+x**3-3*x**2-x+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}} \,d x } \]

[In]

integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x, algorithm="maxima")

[Out]

integrate((x^4 - 3*x^2 + 1)*(x^2 + 1)/((x^4 + x^3 - 3*x^2 - x + 1)*x^2*sqrt((2*x^2 + x - 2)/(x^2 + x - 1))), x
)

Giac [F]

\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}} \,d x } \]

[In]

integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2-x+1),x, algorithm="giac")

[Out]

integrate((x^4 - 3*x^2 + 1)*(x^2 + 1)/((x^4 + x^3 - 3*x^2 - x + 1)*x^2*sqrt((2*x^2 + x - 2)/(x^2 + x - 1))), x
)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int \frac {\left (x^2+1\right )\,\left (x^4-3\,x^2+1\right )}{x^2\,\sqrt {\frac {2\,x^2+x-2}{x^2+x-1}}\,\left (x^4+x^3-3\,x^2-x+1\right )} \,d x \]

[In]

int(((x^2 + 1)*(x^4 - 3*x^2 + 1))/(x^2*((x + 2*x^2 - 2)/(x + x^2 - 1))^(1/2)*(x^3 - 3*x^2 - x + x^4 + 1)),x)

[Out]

int(((x^2 + 1)*(x^4 - 3*x^2 + 1))/(x^2*((x + 2*x^2 - 2)/(x + x^2 - 1))^(1/2)*(x^3 - 3*x^2 - x + x^4 + 1)), x)