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

Optimal. Leaf size=98 \[ \frac {\left (1+2 x+x^2\right ) \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )+2 \log (x)-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}\right ) \]

[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))

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Rubi [F]
time = 4.66, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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 (\tan ^{-1}\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 ) F\left (\tan ^{-1}\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 (\tan ^{-1}\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 ) F\left (\tan ^{-1}\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} \tanh ^{-1}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\tanh ^{-1}\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 (\tan ^{-1}\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 ) F\left (\tan ^{-1}\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 )}\\ &=-\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\frac {\left (3-i \sqrt {3}\right ) x \left (3+\sqrt {5}+2 x^2\right )}{6 \sqrt {1+3 x^2+x^4}}+\frac {\left (3+i \sqrt {3}\right ) x \left (3+\sqrt {5}+2 x^2\right )}{6 \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} \tanh ^{-1}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {2+3 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )+\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {1}{6} \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 (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {1+3 x^2+x^4}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {\frac {1}{6} \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 (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {1+3 x^2+x^4}}+\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 (\tan ^{-1}\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 {\left (2 i-\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\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 {\left (2 i+\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\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 {2 \left (i+\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (2-i \sqrt {3}-\sqrt {5}\right ) \sqrt {1+3 x^2+x^4}}-\frac {2 \left (i-\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (2+i \sqrt {3}-\sqrt {5}\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 ) F\left (\tan ^{-1}\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 {(8 i) \text {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )}{\sqrt {3}}+\frac {(8 i) \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+3 x+x^2}} \, dx,x,x^2\right )}{\sqrt {3}}-\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 (-9+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+3 x+x^2}} \, dx,x,x^2\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}{12} \left (9+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+3 x+x^2}} \, dx,x,x^2\right )+\frac {\left (8 \left (3 i+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {5}}+\frac {x^2}{2}} \sqrt {3-\sqrt {5}+2 x^2}\right ) \int \frac {\sqrt {3-\sqrt {5}+2 x^2}}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {\frac {1}{3-\sqrt {5}}+\frac {x^2}{2}}} \, dx}{3 \left (2 i-\sqrt {3}-i \sqrt {5}\right ) \sqrt {1+3 x^2+x^4}}+\frac {\left (8 \left (3 i-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {5}}+\frac {x^2}{2}} \sqrt {3-\sqrt {5}+2 x^2}\right ) \int \frac {\sqrt {3-\sqrt {5}+2 x^2}}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {\frac {1}{3-\sqrt {5}}+\frac {x^2}{2}}} \, dx}{3 \left (2 i+\sqrt {3}-i \sqrt {5}\right ) \sqrt {1+3 x^2+x^4}}\\ &=-\frac {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\frac {\left (3-i \sqrt {3}\right ) x \left (3+\sqrt {5}+2 x^2\right )}{6 \sqrt {1+3 x^2+x^4}}+\frac {\left (3+i \sqrt {3}\right ) x \left (3+\sqrt {5}+2 x^2\right )}{6 \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} \tanh ^{-1}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {2+3 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )+\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {1}{6} \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 (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {1+3 x^2+x^4}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {\frac {1}{6} \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 (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {1+3 x^2+x^4}}+\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 (\tan ^{-1}\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 {\left (2 i-\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\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 {\left (2 i+\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\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 {2 \left (i+\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (2-i \sqrt {3}-\sqrt {5}\right ) \sqrt {1+3 x^2+x^4}}-\frac {2 \left (i-\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (2+i \sqrt {3}-\sqrt {5}\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 ) F\left (\tan ^{-1}\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 {8 \sqrt {\frac {2}{3} \left (9-4 \sqrt {5}\right )} \sqrt {\frac {2}{3-\sqrt {5}}+x^2} \sqrt {3+\sqrt {5}+2 x^2} \Pi \left (1-\frac {2 \left (3-\sqrt {5}\right )}{\left (i-\sqrt {3}\right )^2};\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (i+3 \sqrt {3}+i \sqrt {5}-\sqrt {15}\right ) \sqrt {\frac {3+\sqrt {5}+2 x^2}{3-\sqrt {5}+2 x^2}} \sqrt {1+3 x^2+x^4}}+\frac {8 \sqrt {\frac {2}{3} \left (9-4 \sqrt {5}\right )} \sqrt {\frac {2}{3-\sqrt {5}}+x^2} \sqrt {3+\sqrt {5}+2 x^2} \Pi \left (1-\frac {2 \left (3-\sqrt {5}\right )}{\left (i+\sqrt {3}\right )^2};\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (i-3 \sqrt {3}+i \sqrt {5}+\sqrt {15}\right ) \sqrt {\frac {3+\sqrt {5}+2 x^2}{3-\sqrt {5}+2 x^2}} \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 {(16 i) \text {Subst}\left (\int \frac {1}{64+48 \left (1-i \sqrt {3}\right )^2+4 \left (1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-8-3 \left (1-i \sqrt {3}\right )^2-4 \left (2-i \sqrt {3}\right ) x^2}{\sqrt {1+3 x^2+x^4}}\right )}{\sqrt {3}}-\frac {(16 i) \text {Subst}\left (\int \frac {1}{64+48 \left (1+i \sqrt {3}\right )^2+4 \left (1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-8-3 \left (1+i \sqrt {3}\right )^2-4 \left (2+i \sqrt {3}\right ) x^2}{\sqrt {1+3 x^2+x^4}}\right )}{\sqrt {3}}-\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}{6} \left (-9+i \sqrt {3}\right ) \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}{6} \left (9+i \sqrt {3}\right ) \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 {x \left (3+\sqrt {5}+2 x^2\right )}{\sqrt {1+3 x^2+x^4}}+\frac {\left (3-i \sqrt {3}\right ) x \left (3+\sqrt {5}+2 x^2\right )}{6 \sqrt {1+3 x^2+x^4}}+\frac {\left (3+i \sqrt {3}\right ) x \left (3+\sqrt {5}+2 x^2\right )}{6 \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 {2 i \tan ^{-1}\left (\frac {1-3 i \sqrt {3}+2 \left (2-i \sqrt {3}\right ) x^2}{4 \sqrt {1+i \sqrt {3}} \sqrt {1+3 x^2+x^4}}\right )}{\sqrt {3 \left (1+i \sqrt {3}\right )}}-\frac {2 i \tan ^{-1}\left (\frac {1+3 i \sqrt {3}+2 \left (2+i \sqrt {3}\right ) x^2}{4 \sqrt {1-i \sqrt {3}} \sqrt {1+3 x^2+x^4}}\right )}{\sqrt {3 \left (1-i \sqrt {3}\right )}}+\frac {3}{2} \tanh ^{-1}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\frac {1}{12} \left (9-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\frac {1}{12} \left (9+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {3+2 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {2+3 x^2}{2 \sqrt {1+3 x^2+x^4}}\right )+\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {1}{6} \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 (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {1+3 x^2+x^4}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {\frac {1}{6} \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 (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {1+3 x^2+x^4}}+\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 (\tan ^{-1}\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 {\left (2 i-\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\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 {\left (2 i+\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\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 {2 \left (i+\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (2-i \sqrt {3}-\sqrt {5}\right ) \sqrt {1+3 x^2+x^4}}-\frac {2 \left (i-\sqrt {3}\right ) \sqrt {\frac {2}{3 \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 ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (2+i \sqrt {3}-\sqrt {5}\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 ) F\left (\tan ^{-1}\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 {8 \sqrt {\frac {2}{3} \left (9-4 \sqrt {5}\right )} \sqrt {\frac {2}{3-\sqrt {5}}+x^2} \sqrt {3+\sqrt {5}+2 x^2} \Pi \left (1-\frac {2 \left (3-\sqrt {5}\right )}{\left (i-\sqrt {3}\right )^2};\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (i+3 \sqrt {3}+i \sqrt {5}-\sqrt {15}\right ) \sqrt {\frac {3+\sqrt {5}+2 x^2}{3-\sqrt {5}+2 x^2}} \sqrt {1+3 x^2+x^4}}+\frac {8 \sqrt {\frac {2}{3} \left (9-4 \sqrt {5}\right )} \sqrt {\frac {2}{3-\sqrt {5}}+x^2} \sqrt {3+\sqrt {5}+2 x^2} \Pi \left (1-\frac {2 \left (3-\sqrt {5}\right )}{\left (i+\sqrt {3}\right )^2};\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\left (i-3 \sqrt {3}+i \sqrt {5}+\sqrt {15}\right ) \sqrt {\frac {3+\sqrt {5}+2 x^2}{3-\sqrt {5}+2 x^2}} \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}{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\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 94, normalized size = 0.96 \begin {gather*} \frac {(1+x)^2 \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-2 \tanh ^{-1}\left (\frac {\sqrt {1+3 x^2+x^4}}{1+x^2}\right )-3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.56, size = 911, normalized size = 9.30 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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]

1/x*(x^4+3*x^2+1)^(1/2)+3/(1/2*I*5^(1/2)-1/2*I)*(1-(1/2*5^(1/2)-3/2)*x^2)^(1/2)*(1-(-3/2-1/2*5^(1/2))*x^2)^(1/
2)/(x^4+3*x^2+1)^(1/2)*EllipticF(x*(1/2*I*5^(1/2)-1/2*I),3/2+1/2*5^(1/2))+(x^4+3*x^2+1)^(1/2)/(x^2+x+1)-5/2*ln
(2*x^2+3+2*(x^4+3*x^2+1)^(1/2))-(3/2-1/6*I*3^(1/2))*(1/2/(-1-I*3^(1/2))^(1/2)*arctanh(1/14*(-2+I*3^(1/2))*(7*x
^2+11/2-5/2*I*3^(1/2))/(-1-I*3^(1/2))^(1/2)/(x^4+3*x^2+1)^(1/2))-1/(1/2*5^(1/2)-3/2)^(1/2)*(-1/2-1/2*I*3^(1/2)
)*(1-(1/2*5^(1/2)-3/2)*x^2)^(1/2)*(1-(-3/2-1/2*5^(1/2))*x^2)^(1/2)/(x^4+3*x^2+1)^(1/2)*EllipticPi((1/2*5^(1/2)
-3/2)^(1/2)*x,-1/2*(-1/2+1/2*I*3^(1/2))*5^(1/2)+3/4-3/4*I*3^(1/2),(-3/2-1/2*5^(1/2))^(1/2)/(1/2*5^(1/2)-3/2)^(
1/2)))-(3/2+1/6*I*3^(1/2))*(1/2/(-1+I*3^(1/2))^(1/2)*arctanh(1/14*(-2-I*3^(1/2))*(7*x^2+11/2+5/2*I*3^(1/2))/(-
1+I*3^(1/2))^(1/2)/(x^4+3*x^2+1)^(1/2))-1/(1/2*5^(1/2)-3/2)^(1/2)*(-1/2+1/2*I*3^(1/2))*(1-(1/2*5^(1/2)-3/2)*x^
2)^(1/2)*(1-(-3/2-1/2*5^(1/2))*x^2)^(1/2)/(x^4+3*x^2+1)^(1/2)*EllipticPi((1/2*5^(1/2)-3/2)^(1/2)*x,-1/2*(-1/2-
1/2*I*3^(1/2))*5^(1/2)+3/4+3/4*I*3^(1/2),(-3/2-1/2*5^(1/2))^(1/2)/(1/2*5^(1/2)-3/2)^(1/2)))+4/3*I*3^(1/2)*(1/2
/(-1-I*3^(1/2))^(1/2)*arctanh(1/14*(-2+I*3^(1/2))*(7*x^2+11/2-5/2*I*3^(1/2))/(-1-I*3^(1/2))^(1/2)/(x^4+3*x^2+1
)^(1/2))-1/(1/2*5^(1/2)-3/2)^(1/2)*(-1/2-1/2*I*3^(1/2))*(1-(1/2*5^(1/2)-3/2)*x^2)^(1/2)*(1-(-3/2-1/2*5^(1/2))*
x^2)^(1/2)/(x^4+3*x^2+1)^(1/2)*EllipticPi((1/2*5^(1/2)-3/2)^(1/2)*x,-1/2*(-1/2+1/2*I*3^(1/2))*5^(1/2)+3/4-3/4*
I*3^(1/2),(-3/2-1/2*5^(1/2))^(1/2)/(1/2*5^(1/2)-3/2)^(1/2)))-4/3*I*3^(1/2)*(1/2/(-1+I*3^(1/2))^(1/2)*arctanh(1
/14*(-2-I*3^(1/2))*(7*x^2+11/2+5/2*I*3^(1/2))/(-1+I*3^(1/2))^(1/2)/(x^4+3*x^2+1)^(1/2))-1/(1/2*5^(1/2)-3/2)^(1
/2)*(-1/2+1/2*I*3^(1/2))*(1-(1/2*5^(1/2)-3/2)*x^2)^(1/2)*(1-(-3/2-1/2*5^(1/2))*x^2)^(1/2)/(x^4+3*x^2+1)^(1/2)*
EllipticPi((1/2*5^(1/2)-3/2)^(1/2)*x,-1/2*(-1/2-1/2*I*3^(1/2))*5^(1/2)+3/4+3/4*I*3^(1/2),(-3/2-1/2*5^(1/2))^(1
/2)/(1/2*5^(1/2)-3/2)^(1/2)))+3/2*ln(3/2+x^2+(x^4+3*x^2+1)^(1/2))-arctanh(1/2*(3*x^2+2)/(x^4+3*x^2+1)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.44, size = 150, normalized size = 1.53 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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