\(\int \frac {(-1+x^2) (1+x^2) \sqrt {1+3 x^2+x^4}}{x^2 (1+x+x^2)^2} \, dx\) [1358]

Optimal result

Integrand size = 36, antiderivative size = 98 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {\left (1+2 x+x^2\right ) \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )+2 \log (x)-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

-((x*(3 + Sqrt[5] + 2*x^2))/Sqrt[1 + 3*x^2 + x^4]) + ((3 - I*Sqrt[3])*x*(3 + Sqrt[5] + 2*x^2))/(6*Sqrt[1 + 3*x
^2 + x^4]) + ((3 + I*Sqrt[3])*x*(3 + Sqrt[5] + 2*x^2))/(6*Sqrt[1 + 3*x^2 + x^4]) + Sqrt[1 + 3*x^2 + x^4] - ((3
- I*Sqrt[3])*Sqrt[1 + 3*x^2 + x^4])/6 - ((3 + I*Sqrt[3])*Sqrt[1 + 3*x^2 + x^4])/6 + Sqrt[1 + 3*x^2 + x^4]/x +
(Sqrt[(1 + I*Sqrt[3])/3]*(I + Sqrt[3])*ArcTan[(1 - (3*I)*Sqrt[3] + 2*(2 - I*Sqrt[3])*x^2)/(4*Sqrt[1 + I*Sqrt[
3]]*Sqrt[1 + 3*x^2 + x^4])])/2 - ((I - Sqrt[3])*Sqrt[(1 - I*Sqrt[3])/3]*ArcTan[(1 + (3*I)*Sqrt[3] + 2*(2 + I*S
qrt[3])*x^2)/(4*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/2 + (3*ArcTanh[(3 + 2*x^2)/(2*Sqrt[1 + 3*x^2 + x^
4])])/2 - ((9 - I*Sqrt[3])*ArcTanh[(3 + 2*x^2)/(2*Sqrt[1 + 3*x^2 + x^4])])/12 - ((9 + I*Sqrt[3])*ArcTanh[(3 +
2*x^2)/(2*Sqrt[1 + 3*x^2 + x^4])])/12 - ArcTanh[(2 + 3*x^2)/(2*Sqrt[1 + 3*x^2 + x^4])] + ((I - Sqrt[3])*Sqrt[(
3 + Sqrt[5])/6]*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticE[ArcTan
[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(2*Sqrt[1 + 3*x^2 + x^4]) - ((I + Sqrt[3])*Sqrt[(3 + Sqrt[5])/
6]*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticE[ArcTan[Sqrt[(3 + Sq
rt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(2*Sqrt[1 + 3*x^2 + x^4]) + (Sqrt[(3 + Sqrt[5])/2]*Sqrt[(2 + (3 - Sqrt[5])*
x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticE[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[
5])/2])/Sqrt[1 + 3*x^2 + x^4] - ((2*I - Sqrt[3])*Sqrt[2/(3*(3 + Sqrt[5]))]*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (
3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/Sqr
t[1 + 3*x^2 + x^4] + ((2*I + Sqrt[3])*Sqrt[2/(3*(3 + Sqrt[5]))]*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5]
)*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/Sqrt[1 + 3*x^2
+ x^4] + (2*(I + Sqrt[3])*Sqrt[2/(3*(3 + Sqrt[5]))]*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2
+ (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/((2 - I*Sqrt[3] - Sqrt[5]
)*Sqrt[1 + 3*x^2 + x^4]) - (2*(I - Sqrt[3])*Sqrt[2/(3*(3 + Sqrt[5]))]*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + S
qrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/((2 + I*
Sqrt[3] - Sqrt[5])*Sqrt[1 + 3*x^2 + x^4]) - (3*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 +
Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 +
3*x^2 + x^4]) - (4*(1 + I*Sqrt[3])*Sqrt[(9 - 4*Sqrt[5])/3]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[
5]))/(I - Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/((5*I + Sqrt[3] - I*Sqrt[5] - Sqrt
[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) + (4*(1 - I*Sqrt[3])*Sqrt[(9 -
4*Sqrt[5])/3]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I + Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5]
)/2]*x], (-5 + 3*Sqrt[5])/2])/((5*I - Sqrt[3] - I*Sqrt[5] + Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5]
+ 2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) + (4*Defer[Int][Sqrt[1 + 3*x^2 + x^4]/(-1 + I*Sqrt[3] - 2*x)^2, x])/3 - (4*(1
- I*Sqrt[3])*Defer[Int][Sqrt[1 + 3*x^2 + x^4]/(-1 + I*Sqrt[3] - 2*x)^2, x])/3 + (4*Defer[Int][Sqrt[1 + 3*x^2
+ x^4]/(1 + I*Sqrt[3] + 2*x)^2, x])/3 - (4*(1 + I*Sqrt[3])*Defer[Int][Sqrt[1 + 3*x^2 + x^4]/(1 + I*Sqrt[3] + 2
*x)^2, x])/3

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {1+3 x^2+x^4}}{x^2}+\frac {2 \sqrt {1+3 x^2+x^4}}{x}+\frac {(-1-2 x) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2}-\frac {2 x \sqrt {1+3 x^2+x^4}}{1+x+x^2}\right ) \, dx \\ & = 2 \int \frac {\sqrt {1+3 x^2+x^4}}{x} \, dx-2 \int \frac {x \sqrt {1+3 x^2+x^4}}{1+x+x^2} \, dx-\int \frac {\sqrt {1+3 x^2+x^4}}{x^2} \, dx+\int \frac {(-1-2 x) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2} \, dx \\ & = \frac {\sqrt {1+3 x^2+x^4}}{x}-2 \int \left (\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt {1+3 x^2+x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt {1+3 x^2+x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx-\int \frac {3+2 x^2}{\sqrt {1+3 x^2+x^4}} \, dx+\int \left (-\frac {\sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2}-\frac {2 x \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2}\right ) \, dx+\text {Subst}\left (\int \frac {\sqrt {1+3 x+x^2}}{x} \, dx,x,x^2\right ) \\ & = \sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}-\frac {1}{2} \text {Subst}\left (\int \frac {-2-3 x}{x \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )-2 \int \frac {x^2}{\sqrt {1+3 x^2+x^4}} \, dx-2 \int \frac {x \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2} \, dx-3 \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{3} \left (2 \left (3-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{1+i \sqrt {3}+2 x} \, dx-\frac {1}{3} \left (2 \left (3+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{1-i \sqrt {3}+2 x} \, dx-\int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+x+x^2\right )^2} \, dx \\ & = -\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}+\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {1+3 x^2+x^4}}-\frac {3 \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1+3 x^2+x^4}}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+3 x+x^2}} \, dx,x,x^2\right )-2 \int \left (-\frac {2 \left (-1+i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}-\frac {2 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}-\frac {2 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx-\frac {1}{3} \left (4 \left (3-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{3} \left (4 \left (3-i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{3} \left (4 \left (3+i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+3 x^2+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{3} \left (4 \left (3+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx-\int \left (-\frac {4 \sqrt {1+3 x^2+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 \sqrt {1+3 x^2+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \sqrt {1+3 x^2+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x \sqrt {1+3 x+x^2}} \, dx,x,x^2\right ) \\ & = -\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}+\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {1+3 x^2+x^4}}-\frac {3 \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1+3 x^2+x^4}}+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+3 x^2}{\sqrt {1+3 x^2+x^4}}\right )+3 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {3+2 x^2}{\sqrt {1+3 x^2+x^4}}\right )-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{3} \left (2 \left (3-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {1+3 x+x^2}}{\left (1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{3} \left (8 \left (3-i \sqrt {3}\right )\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{12} \left (-3+i \sqrt {3}\right ) \int \frac {12+\left (1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {12+\left (1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{3} \left (2 \left (3+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {1+3 x+x^2}}{\left (1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{3} \left (8 \left (3+i \sqrt {3}\right )\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx \\ & = -\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\sqrt {1+3 x^2+x^4}-\frac {1}{6} \left (3-i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}-\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt {1+3 x^2+x^4}+\frac {\sqrt {1+3 x^2+x^4}}{x}+\frac {3}{2} \text {arctanh}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\text {arctanh}\left (\frac {2+3 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )+\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {1+3 x^2+x^4}}-\frac {3 \sqrt {\frac {2+\left (3-\sqrt {5}\right ) x^2}{2+\left (3+\sqrt {5}\right ) x^2}} \left (2+\left (3+\sqrt {5}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1+3 x^2+x^4}}+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {4}{3} \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{12} \left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {2 \left (1+3 i \sqrt {3}\right )+4 \left (2+i \sqrt {3}\right ) x}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (-3+i \sqrt {3}\right ) \int \frac {x^2}{\sqrt {1+3 x^2+x^4}} \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+3 x^2+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {2 \left (1-3 i \sqrt {3}\right )+4 \left (2-i \sqrt {3}\right ) x}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (3+i \sqrt {3}\right ) \int \frac {x^2}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{3} \left (2 \left (3-2 i \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{3} \left (2 \left (3+2 i \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx+\frac {\left (4 \left (3 i+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i-\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (8 \left (3 i+\sqrt {3}\right )\right ) \int \frac {3-\sqrt {5}+2 x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i-\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (4 \left (3 i-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i+\sqrt {3}-i \sqrt {5}\right )}+\frac {\left (8 \left (3 i-\sqrt {3}\right )\right ) \int \frac {3-\sqrt {5}+2 x^2}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx}{3 \left (2 i+\sqrt {3}-i \sqrt {5}\right )} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {(1+x)^2 \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )+2 \log (x)-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}\right ) \]

[In]

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

[Out]

((1 + x)^2*Sqrt[1 + 3*x^2 + x^4])/(x*(1 + x + x^2)) - 3*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 + x + x^2 + Sqrt[1 + 3*
x^2 + x^4])] + 2*Log[x] - 2*Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4]]

Maple [A] (verified)

Time = 6.48 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}-2 \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right )+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )}{2}\)	\(77\)
default	\(\frac {3 x \sqrt {2}\, \left (x^{2}+x +1\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+2 \left (1+x \right )^{2} \sqrt {x^{4}+3 x^{2}+1}-4 x \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right ) \left (x^{2}+x +1\right )}{2 x \left (x^{2}+x +1\right )}\)	\(91\)
pseudoelliptic	\(\frac {3 x \sqrt {2}\, \left (x^{2}+x +1\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+2 \left (1+x \right )^{2} \sqrt {x^{4}+3 x^{2}+1}-4 x \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right ) \left (x^{2}+x +1\right )}{2 x \left (x^{2}+x +1\right )}\)	\(91\)
trager	\(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +2 \sqrt {x^{4}+3 x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}+x +1}\right )}{2}-2 \ln \left (\frac {1+x^{2}+\sqrt {x^{4}+3 x^{2}+1}}{x}\right )\)	\(118\)
elliptic	\(-\ln \left (x^{2}+\frac {3}{2}+\sqrt {x^{4}+3 x^{2}+1}\right )-\operatorname {arctanh}\left (\frac {3 x^{2}+2}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (\frac {2 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+6 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )-9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )+12 \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\right )}{6 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}\, \left (\frac {x^{2}-1}{-x^{2}-1}+1\right ) \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+3\right )}-\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (\sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )-6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )\right )}{3 \left (\frac {x^{2}-1}{-x^{2}-1}+1\right ) \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}}+\frac {\left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}-\frac {1}{2 \left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}-1\right )}+\frac {3 \ln \left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}-1\right )}{2}-\frac {1}{2 \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}-\frac {3 \ln \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{2}\right ) \sqrt {2}}{2}\)	\(724\)

[In]

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

[Out]

(x^2+2*x+1)*(x^4+3*x^2+1)^(1/2)/x/(x^2+x+1)-2*arcsinh(1/x*(x^2+1))+3/2*2^(1/2)*arctanh(1/2*(x^2-x+1)*2^(1/2)/(
x^4+3*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.53 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {3 \, \sqrt {2} {\left (x^{3} + x^{2} + x\right )} \log \left (\frac {3 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} - x + 1\right )} + 9 \, x^{2} - 2 \, x + 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + 8 \, {\left (x^{3} + x^{2} + x\right )} \log \left (-\frac {x^{2} - \sqrt {x^{4} + 3 \, x^{2} + 1} + 1}{x}\right ) + 4 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 2 \, x + 1\right )}}{4 \, {\left (x^{3} + x^{2} + x\right )}} \]

[In]

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

[Out]

1/4*(3*sqrt(2)*(x^3 + x^2 + x)*log((3*x^4 - 2*x^3 + 2*sqrt(2)*sqrt(x^4 + 3*x^2 + 1)*(x^2 - x + 1) + 9*x^2 - 2*
x + 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) + 8*(x^3 + x^2 + x)*log(-(x^2 - sqrt(x^4 + 3*x^2 + 1) + 1)/x) + 4*sqrt
(x^4 + 3*x^2 + 1)*(x^2 + 2*x + 1))/(x^3 + x^2 + x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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