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

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

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Rubi [F]  time = 6.89, 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+\operatorname {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} \operatorname {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} \operatorname {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+\operatorname {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 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2+3 x^2}{\sqrt {1+3 x^2+x^4}}\right )+3 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [C]  time = 3.86, size = 880, normalized size = 8.98 \begin {gather*} \frac {1}{2} \left (2 \sqrt {x^4+3 x^2+1} \left (\frac {1}{x^2+x+1}+\frac {1}{x}\right )+\frac {3 i \left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (4-2 i \sqrt {3}\right ) x^2-3 i \sqrt {3}+1}{4 \sqrt {1+i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )}{2 \sqrt {1+i \sqrt {3}}}-\frac {3 \left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (4+2 i \sqrt {3}\right ) x^2+3 i \sqrt {3}+1}{4 \sqrt {1-i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )}{2 \sqrt {1-i \sqrt {3}}}-2 \tanh ^{-1}\left (\frac {2 x^2+3}{2 \sqrt {x^4+3 x^2+1}}\right )-2 \tanh ^{-1}\left (\frac {3 x^2+2}{2 \sqrt {x^4+3 x^2+1}}\right )-\frac {3 i \sqrt {2} \sqrt {\frac {-2 x^2+\sqrt {5}-3}{-3+\sqrt {5}}} \sqrt {2 x^2+\sqrt {5}+3} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {7}{2}+\frac {3 \sqrt {5}}{2}\right )}{\sqrt {x^4+3 x^2+1}}+\frac {3 (-1)^{5/6} \sqrt {2} \sqrt {\frac {-2 x^2+\sqrt {5}-3}{-3+\sqrt {5}}} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (\frac {1}{2} \sqrt [3]{-1} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+3 x^2+1}}+\frac {3 i \sqrt {2} \sqrt {\frac {-2 x^2+\sqrt {5}-3}{-3+\sqrt {5}}} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (\frac {1}{2} \sqrt [3]{-1} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+3 x^2+1}}+\frac {3 (-1)^{5/6} \sqrt {2} \sqrt {\frac {-2 x^2+\sqrt {5}-3}{-3+\sqrt {5}}} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (-\frac {1}{2} (-1)^{2/3} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+3 x^2+1}}+\frac {3 i \sqrt {2} \sqrt {\frac {-2 x^2+\sqrt {5}-3}{-3+\sqrt {5}}} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (-\frac {1}{2} (-1)^{2/3} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+3 x^2+1}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[1 + 3*x^2 + x^4]*(x^(-1) + (1 + x + x^2)^(-1)) + (((3*I)/2)*(I + Sqrt[3])*ArcTan[(1 - (3*I)*Sqrt[3] +
(4 - (2*I)*Sqrt[3])*x^2)/(4*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/Sqrt[1 + I*Sqrt[3]] - (3*(1 + I*Sqrt[
3])*ArcTan[(1 + (3*I)*Sqrt[3] + (4 + (2*I)*Sqrt[3])*x^2)/(4*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/(2*Sq
rt[1 - I*Sqrt[3]]) - 2*ArcTanh[(3 + 2*x^2)/(2*Sqrt[1 + 3*x^2 + x^4])] - 2*ArcTanh[(2 + 3*x^2)/(2*Sqrt[1 + 3*x^
2 + x^4])] - ((3*I)*Sqrt[2]*Sqrt[(-3 + Sqrt[5] - 2*x^2)/(-3 + Sqrt[5])]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticF[I*
ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], 7/2 + (3*Sqrt[5])/2])/Sqrt[1 + 3*x^2 + x^4] + ((3*I)*Sqrt[2]*Sqrt[(-3 + Sqrt
[5] - 2*x^2)/(-3 + Sqrt[5])]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[((-1)^(1/3)*(3 + Sqrt[5]))/2, I*ArcSinh[Sqrt
[2/(3 + Sqrt[5])]*x], (7 + 3*Sqrt[5])/2])/((1 + (-1)^(1/3))*Sqrt[1 + 3*x^2 + x^4]) + (3*(-1)^(5/6)*Sqrt[2]*Sqr
t[(-3 + Sqrt[5] - 2*x^2)/(-3 + Sqrt[5])]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[((-1)^(1/3)*(3 + Sqrt[5]))/2, I*
ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], (7 + 3*Sqrt[5])/2])/((1 + (-1)^(1/3))*Sqrt[1 + 3*x^2 + x^4]) + ((3*I)*Sqrt[2
]*Sqrt[(-3 + Sqrt[5] - 2*x^2)/(-3 + Sqrt[5])]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[-1/2*((-1)^(2/3)*(3 + Sqrt[
5])), I*ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], (7 + 3*Sqrt[5])/2])/((1 + (-1)^(1/3))*Sqrt[1 + 3*x^2 + x^4]) + (3*(-
1)^(5/6)*Sqrt[2]*Sqrt[(-3 + Sqrt[5] - 2*x^2)/(-3 + Sqrt[5])]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[-1/2*((-1)^(
2/3)*(3 + Sqrt[5])), I*ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], (7 + 3*Sqrt[5])/2])/((1 + (-1)^(1/3))*Sqrt[1 + 3*x^2
+ x^4]))/2

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IntegrateAlgebraic [A]  time = 1.07, size = 98, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*x + 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]]

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fricas [A]  time = 0.64, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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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maple [C]  time = 0.90, size = 117, normalized size = 1.19

method	result	size
trager	\(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}-\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\RootOf \left (\textit {\_Z}^{2}-2\right )-2 \sqrt {x^{4}+3 x^{2}+1}}{x^{2}+x +1}\right )}{2}-2 \ln \left (\frac {1+x^{2}+\sqrt {x^{4}+3 x^{2}+1}}{x}\right )\)	\(117\)
risch	\(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}+\frac {3 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{\left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}-\ln \left (\frac {3}{2}+x^{2}+\sqrt {x^{4}+3 x^{2}+1}\right )-\arctanh \left (\frac {3 x^{2}+2}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )-\frac {3 \arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4 \sqrt {-1-i \sqrt {3}}}-\frac {3 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}+\frac {3 i \sqrt {3}\, \arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4 \sqrt {-1-i \sqrt {3}}}-\frac {3 \arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4 \sqrt {-1+i \sqrt {3}}}-\frac {3 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}-\frac {3 i \sqrt {3}\, \arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4 \sqrt {-1+i \sqrt {3}}}\)	\(608\)
elliptic	\(-\ln \left (\frac {3}{2}+x^{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 {3 \sqrt {2}\, \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}}+\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}}-9 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )+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 )+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 )}-\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 (6 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )-\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 )\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 (-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}-\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}\)	\(725\)
default	\(\frac {\sqrt {x^{4}+3 x^{2}+1}}{x}+\frac {3 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{\left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}+\frac {3 \ln \left (\frac {3}{2}+x^{2}+\sqrt {x^{4}+3 x^{2}+1}\right )}{2}-\arctanh \left (\frac {3 x^{2}+2}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )-\frac {5 \ln \left (2 x^{2}+3+2 \sqrt {x^{4}+3 x^{2}+1}\right )}{2}+\frac {4 i \sqrt {3}\, \left (\frac {\arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{3}-\frac {4 i \sqrt {3}\, \left (\frac {\arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{3}+\frac {\sqrt {x^{4}+3 x^{2}+1}}{x^{2}+x +1}-\left (\frac {3}{2}-\frac {i \sqrt {3}}{6}\right ) \left (\frac {\arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )-\left (\frac {3}{2}+\frac {i \sqrt {3}}{6}\right ) \left (\frac {\arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )\)	\(911\)

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]

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

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