\(\int \frac {(-1-x-x^2+x^4) (2+x+2 x^4)}{\sqrt {-1-x+x^4} (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8)} \, dx\) [1819]

Optimal result

Integrand size = 69, antiderivative size = 123 \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=-(-1)^{3/4} \sqrt {\frac {1}{10} \left (-7 i+\sqrt {15}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {15}}{2}} x}{2 \sqrt {-1-x+x^4}}\right )+\frac {\sqrt [4]{-1} \sqrt {7 i+\sqrt {15}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {15}}{2}} x}{2 \sqrt {-1-x+x^4}}\right )}{\sqrt {10}} \]

[Out]

-1/10*(-1)^(3/4)*(-70*I+10*15^(1/2))^(1/2)*arctan(1/4*(2-2*I*15^(1/2))^(1/2)*x/(x^4-x-1)^(1/2))+1/10*(-1)^(1/4
)*(7*I+15^(1/2))^(1/2)*arctan(1/4*(2+2*I*15^(1/2))^(1/2)*x/(x^4-x-1)^(1/2))*10^(1/2)

Rubi [F]

\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx \]

[In]

Int[((-1 - x - x^2 + x^4)*(2 + x + 2*x^4))/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 +
4*x^8)),x]

[Out]

Defer[Int][1/Sqrt[-1 - x + x^4], x]/2 - 4*Defer[Int][1/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*
x^5 + x^6 + 4*x^8)), x] - 7*Defer[Int][x/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*
x^8)), x] - (9*Defer[Int][x^2/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x])/
2 - Defer[Int][x^3/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x]/2 + (7*Defer
[Int][x^4/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x])/2 + 3*Defer[Int][x^5
/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x] - (5*Defer[Int][x^6/(Sqrt[-1 -
x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x^8)), x])/2

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 \sqrt {-1-x+x^4}}-\frac {8+14 x+9 x^2+x^3-7 x^4-6 x^5+5 x^6}{2 \sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \frac {8+14 x+9 x^2+x^3-7 x^4-6 x^5+5 x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx \\ & = \frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \left (\frac {8}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {14 x}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {9 x^2}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {x^3}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}-\frac {7 x^4}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}-\frac {6 x^5}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {5 x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \frac {x^3}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-\frac {5}{2} \int \frac {x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx+3 \int \frac {x^5}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx+\frac {7}{2} \int \frac {x^4}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-4 \int \frac {1}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-\frac {9}{2} \int \frac {x^2}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-7 \int \frac {x}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\frac {1}{10} \left ((1-i) \sqrt {5 \left (-7 i+\sqrt {15}\right )} \arctan \left (\frac {\sqrt {\frac {1}{8}-\frac {i \sqrt {15}}{8}} x}{\sqrt {-1-x+x^4}}\right )+(1+i) \sqrt {5 \left (7 i+\sqrt {15}\right )} \arctan \left (\frac {\sqrt {\frac {1}{8}+\frac {i \sqrt {15}}{8}} x}{\sqrt {-1-x+x^4}}\right )\right ) \]

[In]

Integrate[((-1 - x - x^2 + x^4)*(2 + x + 2*x^4))/(Sqrt[-1 - x + x^4]*(4 + 8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 +
x^6 + 4*x^8)),x]

[Out]

((1 - I)*Sqrt[5*(-7*I + Sqrt[15])]*ArcTan[(Sqrt[1/8 - (I/8)*Sqrt[15]]*x)/Sqrt[-1 - x + x^4]] + (1 + I)*Sqrt[5*
(7*I + Sqrt[15])]*ArcTan[(Sqrt[1/8 + (I/8)*Sqrt[15]]*x)/Sqrt[-1 - x + x^4]])/10

Maple [A] (verified)

Time = 8.75 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {\left (-\ln \left (\frac {2 x^{4}+\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )+\ln \left (\frac {2 x^{4}-\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )\right ) \sqrt {3}}{4}-\frac {\sqrt {5}\, \left (\arctan \left (\frac {\left (x \sqrt {3}+4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )-\arctan \left (\frac {\left (x \sqrt {3}-4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )\right )}{10}\)	\(139\)
pseudoelliptic	\(\frac {\left (-\ln \left (\frac {2 x^{4}+\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )+\ln \left (\frac {2 x^{4}-\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )\right ) \sqrt {3}}{4}-\frac {\sqrt {5}\, \left (\arctan \left (\frac {\left (x \sqrt {3}+4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )-\arctan \left (\frac {\left (x \sqrt {3}-4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )\right )}{10}\)	\(139\)
trager	\(-\operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) \ln \left (\frac {50 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{4} x^{2}+10 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x^{4}-25 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}-3 x^{4}-10 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x -5 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) \sqrt {x^{4}-x -1}\, x -10 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+3 x^{2}+3 x +3}{5 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}-x^{4}-x^{2}+x +1}\right )-5 \ln \left (-\frac {400 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{5} x^{2}-80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{4}+120 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{2}-36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{4}+80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x -160 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} \sqrt {x^{4}-x -1}\, x +80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}-27 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{2}+36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x +56 \sqrt {x^{4}-x -1}\, x +36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{20 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+4 x^{4}-3 x^{2}-4 x -4}\right ) \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}+\frac {7 \ln \left (-\frac {400 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{5} x^{2}-80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{4}+120 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{2}-36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{4}+80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x -160 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} \sqrt {x^{4}-x -1}\, x +80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}-27 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{2}+36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x +56 \sqrt {x^{4}-x -1}\, x +36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{20 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+4 x^{4}-3 x^{2}-4 x -4}\right ) \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{4}\)	\(710\)
elliptic	\(\text {Expression too large to display}\)	\(8121\)

[In]

int((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (76) = 152\).

Time = 0.55 (sec) , antiderivative size = 807, normalized size of antiderivative = 6.56 \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=-\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} \sqrt {-3} + 7} \log \left (-\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} - \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {\sqrt {5} \sqrt {-3} + 7} + 40 \, {\left (7 \, x^{5} - x^{3} + \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} \sqrt {-3} + 7} \log \left (\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} - \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {\sqrt {5} \sqrt {-3} + 7} - 40 \, {\left (7 \, x^{5} - x^{3} + \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} \sqrt {-3} + 7} \log \left (-\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} + \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {-\sqrt {5} \sqrt {-3} + 7} + 40 \, {\left (7 \, x^{5} - x^{3} - \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} \sqrt {-3} + 7} \log \left (\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} + \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {-\sqrt {5} \sqrt {-3} + 7} - 40 \, {\left (7 \, x^{5} - x^{3} - \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) \]

[In]

integrate((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x, algorithm="fric
as")

[Out]

-1/40*sqrt(10)*sqrt(sqrt(5)*sqrt(-3) + 7)*log(-(sqrt(10)*(20*x^8 + 15*x^6 - 40*x^5 - 45*x^4 - 15*x^3 - sqrt(5)
*sqrt(-3)*(4*x^8 - 5*x^6 - 8*x^5 - 9*x^4 + 5*x^3 + 9*x^2 + 8*x + 4) + 5*x^2 + 40*x + 20)*sqrt(sqrt(5)*sqrt(-3)
+ 7) + 40*(7*x^5 - x^3 + sqrt(5)*sqrt(-3)*(x^5 + x^3 - x^2 - x) - 7*x^2 - 7*x)*sqrt(x^4 - x - 1))/(4*x^8 + x^
6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)) + 1/40*sqrt(10)*sqrt(sqrt(5)*sqrt(-3) + 7)*log((sqrt(10)*(20*x^8 +
15*x^6 - 40*x^5 - 45*x^4 - 15*x^3 - sqrt(5)*sqrt(-3)*(4*x^8 - 5*x^6 - 8*x^5 - 9*x^4 + 5*x^3 + 9*x^2 + 8*x + 4
) + 5*x^2 + 40*x + 20)*sqrt(sqrt(5)*sqrt(-3) + 7) - 40*(7*x^5 - x^3 + sqrt(5)*sqrt(-3)*(x^5 + x^3 - x^2 - x) -
7*x^2 - 7*x)*sqrt(x^4 - x - 1))/(4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)) - 1/40*sqrt(10)*sqrt(-
sqrt(5)*sqrt(-3) + 7)*log(-(sqrt(10)*(20*x^8 + 15*x^6 - 40*x^5 - 45*x^4 - 15*x^3 + sqrt(5)*sqrt(-3)*(4*x^8 - 5
*x^6 - 8*x^5 - 9*x^4 + 5*x^3 + 9*x^2 + 8*x + 4) + 5*x^2 + 40*x + 20)*sqrt(-sqrt(5)*sqrt(-3) + 7) + 40*(7*x^5 -
x^3 - sqrt(5)*sqrt(-3)*(x^5 + x^3 - x^2 - x) - 7*x^2 - 7*x)*sqrt(x^4 - x - 1))/(4*x^8 + x^6 - 8*x^5 - 7*x^4 -
x^3 + 3*x^2 + 8*x + 4)) + 1/40*sqrt(10)*sqrt(-sqrt(5)*sqrt(-3) + 7)*log((sqrt(10)*(20*x^8 + 15*x^6 - 40*x^5 -
45*x^4 - 15*x^3 + sqrt(5)*sqrt(-3)*(4*x^8 - 5*x^6 - 8*x^5 - 9*x^4 + 5*x^3 + 9*x^2 + 8*x + 4) + 5*x^2 + 40*x +
20)*sqrt(-sqrt(5)*sqrt(-3) + 7) - 40*(7*x^5 - x^3 - sqrt(5)*sqrt(-3)*(x^5 + x^3 - x^2 - x) - 7*x^2 - 7*x)*sqr
t(x^4 - x - 1))/(4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4))

Sympy [F]

\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int \frac {\left (x + 1\right ) \left (x^{3} - x^{2} - 1\right ) \left (2 x^{4} + x + 2\right )}{\sqrt {x^{4} - x - 1} \cdot \left (4 x^{8} + x^{6} - 8 x^{5} - 7 x^{4} - x^{3} + 3 x^{2} + 8 x + 4\right )}\, dx \]

[In]

integrate((x**4-x**2-x-1)*(2*x**4+x+2)/(x**4-x-1)**(1/2)/(4*x**8+x**6-8*x**5-7*x**4-x**3+3*x**2+8*x+4),x)

[Out]

Integral((x + 1)*(x**3 - x**2 - 1)*(2*x**4 + x + 2)/(sqrt(x**4 - x - 1)*(4*x**8 + x**6 - 8*x**5 - 7*x**4 - x**
3 + 3*x**2 + 8*x + 4)), x)

Maxima [F]

\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} {\left (x^{4} - x^{2} - x - 1\right )}}{{\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} \sqrt {x^{4} - x - 1}} \,d x } \]

[In]

integrate((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x, algorithm="maxi
ma")

[Out]

integrate((2*x^4 + x + 2)*(x^4 - x^2 - x - 1)/((4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)*sqrt(x^4
- x - 1)), x)

Giac [F]

\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} {\left (x^{4} - x^{2} - x - 1\right )}}{{\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} \sqrt {x^{4} - x - 1}} \,d x } \]

[In]

integrate((x^4-x^2-x-1)*(2*x^4+x+2)/(x^4-x-1)^(1/2)/(4*x^8+x^6-8*x^5-7*x^4-x^3+3*x^2+8*x+4),x, algorithm="giac
")

[Out]

integrate((2*x^4 + x + 2)*(x^4 - x^2 - x - 1)/((4*x^8 + x^6 - 8*x^5 - 7*x^4 - x^3 + 3*x^2 + 8*x + 4)*sqrt(x^4
- x - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int -\frac {\left (2\,x^4+x+2\right )\,\left (-x^4+x^2+x+1\right )}{\sqrt {x^4-x-1}\,\left (4\,x^8+x^6-8\,x^5-7\,x^4-x^3+3\,x^2+8\,x+4\right )} \,d x \]

[In]

int(-((x + 2*x^4 + 2)*(x + x^2 - x^4 + 1))/((x^4 - x - 1)^(1/2)*(8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x
^8 + 4)),x)

[Out]

int(-((x + 2*x^4 + 2)*(x + x^2 - x^4 + 1))/((x^4 - x - 1)^(1/2)*(8*x + 3*x^2 - x^3 - 7*x^4 - 8*x^5 + x^6 + 4*x
^8 + 4)), x)