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

Optimal. Leaf size=133 \[ -2 \tan ^{-1}\left (\frac {\sqrt {x^4+x^2+1}}{x^2-x+1}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {x^4+x^2+1}}{\sqrt {2+\sqrt {5}} \left (x^2-x+1\right )}\right )+\sqrt {\frac {1}{5} \left (2 \sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {x^4+x^2+1}}{\sqrt {\sqrt {5}-2} \left (x^2-x+1\right )}\right ) \]

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Rubi [F]  time = 1.00, 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 ) \sqrt {1+x^2+x^4}}{\left (1+x^2\right ) \left (1+x+x^2+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^2+x^4}}{\left (1+x^2\right ) \left (1+x+x^2+x^3+x^4\right )} \, dx &=\int \left (-\frac {2 \sqrt {1+x^2+x^4}}{1+x^2}+\frac {\left (1+2 x+2 x^2\right ) \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^2+x^4}}{1+x^2} \, dx\right )+\int \frac {\left (1+2 x+2 x^2\right ) \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ &=-\left (2 \int \frac {x^2}{\sqrt {1+x^2+x^4}} \, dx\right )-2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \left (\frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}+\frac {2 x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}+\frac {2 x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx\right )+2 \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx-\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-\int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ &=-\frac {2 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=-\frac {2 x \sqrt {1+x^2+x^4}}{1+x^2}-\tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ \end {align*}

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Mathematica [C]  time = 139.37, size = 4461, normalized size = 33.54 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + x^2)*Sqrt[1 + x^2 + x^4]*(1 + x + x^2 + x^3 + x^4)*((-2*Sqrt[1 + x^2 + x^4])/(1 + x^2) + ((1 + 2*x + 2*x
^2)*Sqrt[1 + x^2 + x^4])/(1 + x + x^2 + x^3 + x^4))*(((-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*
x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/Sqrt[1 + x^2 + x^4] + ((-1)^(1/15)*Sqrt[1 + (-1)^(1/3)*x^
2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(2/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 - (-1)^(1/5))*(-
(-1)^(1/5) - (-1)^(2/5))*(-(-1)^(1/5) + (-1)^(3/5))*(-(-1)^(1/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) - ((-1)^(4
/15)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(2/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(
2/3)])/((1 - (-1)^(1/5))*(-(-1)^(1/5) - (-1)^(2/5))*(-(-1)^(1/5) + (-1)^(3/5))*(-(-1)^(1/5) - (-1)^(4/5))*Sqrt
[1 + x^2 + x^4]) + ((-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(2/15), I*Arc
Sinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 - (-1)^(1/5))*(-(-1)^(1/5) - (-1)^(2/5))*(-(-1)^(1/5) + (-1)^(3/5))*(-(-1
)^(1/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) - ((-1)^(13/15)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*E
llipticPi[(-1)^(2/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 - (-1)^(1/5))*(-(-1)^(1/5) - (-1)^(2/5))*(-(-
1)^(1/5) + (-1)^(3/5))*(-(-1)^(1/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) - (2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2
]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((-1 + (-1)^(1/5))*(-1
- (-1)^(2/5))*(-1 + (-1)^(3/5))*(-1 - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) + ((-1)^(1/15)*Sqrt[1 + (-1)^(1/3)*x^2
]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(8/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 + (-1)^(4/5))*((-
1)^(1/5) + (-1)^(4/5))*(-(-1)^(2/5) + (-1)^(4/5))*((-1)^(3/5) + (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) - ((-1)^(4/15
)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(8/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3
)])/((1 + (-1)^(4/5))*((-1)^(1/5) + (-1)^(4/5))*(-(-1)^(2/5) + (-1)^(4/5))*((-1)^(3/5) + (-1)^(4/5))*Sqrt[1 +
x^2 + x^4]) - ((-1)^(7/15)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(8/15), I*ArcSinh
[(-1)^(5/6)*x], (-1)^(2/3)])/((1 + (-1)^(4/5))*((-1)^(1/5) + (-1)^(4/5))*(-(-1)^(2/5) + (-1)^(4/5))*((-1)^(3/5
) + (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) + ((-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticP
i[(-1)^(8/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 + (-1)^(4/5))*((-1)^(1/5) + (-1)^(4/5))*(-(-1)^(2/5)
+ (-1)^(4/5))*((-1)^(3/5) + (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) + ((-1)^(4/15)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 -
(-1)^(2/3)*x^2]*EllipticPi[-(-1)^(11/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 - (-1)^(3/5))*((-1)^(1/5)
- (-1)^(3/5))*(-(-1)^(2/5) - (-1)^(3/5))*(-(-1)^(3/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) + ((-1)^(7/15)*Sqrt[1
+ (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[-(-1)^(11/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((
1 - (-1)^(3/5))*((-1)^(1/5) - (-1)^(3/5))*(-(-1)^(2/5) - (-1)^(3/5))*(-(-1)^(3/5) - (-1)^(4/5))*Sqrt[1 + x^2 +
x^4]) + ((-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[-(-1)^(11/15), I*ArcSinh[(-1
)^(5/6)*x], (-1)^(2/3)])/((1 - (-1)^(3/5))*((-1)^(1/5) - (-1)^(3/5))*(-(-1)^(2/5) - (-1)^(3/5))*(-(-1)^(3/5) -
(-1)^(4/5))*Sqrt[1 + x^2 + x^4]) + ((-1)^(13/15)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi
[-(-1)^(11/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 - (-1)^(3/5))*((-1)^(1/5) - (-1)^(3/5))*(-(-1)^(2/5)
- (-1)^(3/5))*(-(-1)^(3/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) - ((-1)^(1/15)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1
- (-1)^(2/3)*x^2]*EllipticPi[(-1)^(14/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 + (-1)^(2/5))*((-1)^(1/5)
+ (-1)^(2/5))*((-1)^(2/5) + (-1)^(3/5))*((-1)^(2/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) + ((-1)^(7/15)*Sqrt[1
+ (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(14/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1
+ (-1)^(2/5))*((-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + (-1)^(3/5))*((-1)^(2/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4
]) + ((-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(14/15), I*ArcSinh[(-1)^(5/
6)*x], (-1)^(2/3)])/((1 + (-1)^(2/5))*((-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + (-1)^(3/5))*((-1)^(2/5) - (-1)^(
4/5))*Sqrt[1 + x^2 + x^4]) + ((-1)^(13/15)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(
14/15), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/((1 + (-1)^(2/5))*((-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) + (-1)^(
3/5))*((-1)^(2/5) - (-1)^(4/5))*Sqrt[1 + x^2 + x^4]) - ((-1)^(1/5)*((-3 + (-1)^(2/5) - 3*(-1)^(4/5))*Sqrt[6 -
5*(-1)^(1/5) + 5*(-1)^(2/5) - 6*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[(-1)^(1/5) + x^2] + (4 - 5*(-1)^(1/5) + 8*(-1)^
(2/5) - 8*(-1)^(3/5) + 5*(-1)^(4/5))*Sqrt[6 - 6*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[-
(-1)^(2/5) + x^2] + 2*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[(-1)^(3/5) + x^2
] - 2*(-1)^(1/5)*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[(-1)^(3/5) + x^2] - (
-1)^(2/5)*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[(-1)^(3/5) + x^2] + 2*(-1)^(
3/5)*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[(-1)^(3/5) + x^2] - (-1)^(4/5)*Sq
rt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[(-1)^(3/5) + x^2] + 3*Sqrt[6 - 5*(-1)^(1
/5) + 6*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[-(-1)^(4/5) + x^2] - 4*(-1)^(1/5)*Sqrt[6 - 5*(-1)^(1/5)
+ 6*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[-(-1)^(4/5) + x^2] + 3*(-1)^(2/5)*Sqrt[6 - 5*(-1)^(1/5) + 6*
(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[-(-1)^(4/5) + x^2] + 3*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 6*
(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 - (-1)^(1/5) + x^2 - 2*(-1)^(1/5)*x^2 + 2*Sqrt[1 - (-1)^(1/5) + (-1)^(2/5)]*S
qrt[1 + x^2 + x^4]] - (-1)^(2/5)*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 6*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 - (
-1)^(1/5) + x^2 - 2*(-1)^(1/5)*x^2 + 2*Sqrt[1 - (-1)^(1/5) + (-1)^(2/5)]*Sqrt[1 + x^2 + x^4]] + 3*(-1)^(4/5)*S
qrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 6*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 - (-1)^(1/5) + x^2 - 2*(-1)^(1/5)*x^2
+ 2*Sqrt[1 - (-1)^(1/5) + (-1)^(2/5)]*Sqrt[1 + x^2 + x^4]] - 2*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^
(3/5) + 6*(-1)^(4/5)]*Log[2 - (-1)^(3/5) + x^2 - 2*(-1)^(3/5)*x^2 + 2*Sqrt[1 - (-1)^(1/5) - (-1)^(3/5)]*Sqrt[1
+ x^2 + x^4]] + 2*(-1)^(1/5)*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[2 - (-1)
^(3/5) + x^2 - 2*(-1)^(3/5)*x^2 + 2*Sqrt[1 - (-1)^(1/5) - (-1)^(3/5)]*Sqrt[1 + x^2 + x^4]] + (-1)^(2/5)*Sqrt[6
- 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[2 - (-1)^(3/5) + x^2 - 2*(-1)^(3/5)*x^2 + 2*
Sqrt[1 - (-1)^(1/5) - (-1)^(3/5)]*Sqrt[1 + x^2 + x^4]] - 2*(-1)^(3/5)*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5
*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[2 - (-1)^(3/5) + x^2 - 2*(-1)^(3/5)*x^2 + 2*Sqrt[1 - (-1)^(1/5) - (-1)^(3/5)]*
Sqrt[1 + x^2 + x^4]] + (-1)^(4/5)*Sqrt[6 - 5*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 6*(-1)^(4/5)]*Log[2 -
(-1)^(3/5) + x^2 - 2*(-1)^(3/5)*x^2 + 2*Sqrt[1 - (-1)^(1/5) - (-1)^(3/5)]*Sqrt[1 + x^2 + x^4]] - 4*Sqrt[6 - 6*
(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1)^(2/5) + x^2 + 2*(-1)^(2/5)*x^2 + 2*Sqrt[
1 + (-1)^(2/5) + (-1)^(4/5)]*Sqrt[1 + x^2 + x^4]] + 5*(-1)^(1/5)*Sqrt[6 - 6*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)
^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1)^(2/5) + x^2 + 2*(-1)^(2/5)*x^2 + 2*Sqrt[1 + (-1)^(2/5) + (-1)^(4/5)]*Sqrt[
1 + x^2 + x^4]] - 8*(-1)^(2/5)*Sqrt[6 - 6*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1
)^(2/5) + x^2 + 2*(-1)^(2/5)*x^2 + 2*Sqrt[1 + (-1)^(2/5) + (-1)^(4/5)]*Sqrt[1 + x^2 + x^4]] + 8*(-1)^(3/5)*Sqr
t[6 - 6*(-1)^(1/5) + 5*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1)^(2/5) + x^2 + 2*(-1)^(2/5)*x^2 +
2*Sqrt[1 + (-1)^(2/5) + (-1)^(4/5)]*Sqrt[1 + x^2 + x^4]] - 5*(-1)^(4/5)*Sqrt[6 - 6*(-1)^(1/5) + 5*(-1)^(2/5)
- 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1)^(2/5) + x^2 + 2*(-1)^(2/5)*x^2 + 2*Sqrt[1 + (-1)^(2/5) + (-1)^(4/5
)]*Sqrt[1 + x^2 + x^4]] - 3*Sqrt[6 - 5*(-1)^(1/5) + 6*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1)^(
4/5) + x^2 + 2*(-1)^(4/5)*x^2 + 2*Sqrt[1 - (-1)^(3/5) + (-1)^(4/5)]*Sqrt[1 + x^2 + x^4]] + 4*(-1)^(1/5)*Sqrt[6
- 5*(-1)^(1/5) + 6*(-1)^(2/5) - 5*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1)^(4/5) + x^2 + 2*(-1)^(4/5)*x^2 + 2*
Sqrt[1 - (-1)^(3/5) + (-1)^(4/5)]*Sqrt[1 + x^2 + x^4]] - 3*(-1)^(2/5)*Sqrt[6 - 5*(-1)^(1/5) + 6*(-1)^(2/5) - 5
*(-1)^(3/5) + 5*(-1)^(4/5)]*Log[2 + (-1)^(4/5) + x^2 + 2*(-1)^(4/5)*x^2 + 2*Sqrt[1 - (-1)^(3/5) + (-1)^(4/5)]*
Sqrt[1 + x^2 + x^4]]))/(2*(-1 + (-1)^(1/5))^2*(1 + (-1)^(1/5))^4*(1 - (-1)^(1/5) + (-1)^(2/5))^(5/2)*Sqrt[6 -
5*(-1)^(1/5) + 5*(-1)^(2/5) - 6*(-1)^(3/5) + 5*(-1)^(4/5)])))/((-1 + x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2))

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IntegrateAlgebraic [A]  time = 1.06, size = 133, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {1+x^2+x^4}}{1-x+x^2}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-2+\sqrt {5}} \sqrt {1+x^2+x^4}}{1-x+x^2}\right )+\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {5}} \sqrt {1+x^2+x^4}}{1-x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.74, size = 306, normalized size = 2.30 \begin {gather*} \frac {1}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{4} + \sqrt {5} x^{2} + x^{2} + 2\right )} \sqrt {2 \, \sqrt {5} + 2} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} + \sqrt {5} x - x + 2\right )} \sqrt {2 \, \sqrt {5} + 2}}{8 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} + {\left (x^{4} + 3 \, x^{2} + \sqrt {5} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} - {\left (x^{4} + 3 \, x^{2} + \sqrt {5} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*sqrt(2*sqrt(5) + 2)*arctan(1/8*(sqrt(2)*(2*x^4 + sqrt(5)*x^2 + x^2 + 2)*sqrt(2*sqrt(5) + 2)*sqrt(s
qrt(5) + 1) + 2*sqrt(x^4 + x^2 + 1)*(2*x^2 + sqrt(5)*x - x + 2)*sqrt(2*sqrt(5) + 2))/(x^4 - x^3 + x^2 - x + 1)
) + 1/20*sqrt(5)*sqrt(2*sqrt(5) - 2)*log(-(2*sqrt(x^4 + x^2 + 1)*(2*x^2 - sqrt(5)*x + x + 2) + (x^4 + 3*x^2 +
sqrt(5)*(x^4 + x^2 + 1) + 1)*sqrt(2*sqrt(5) - 2))/(x^4 + x^3 + x^2 + x + 1)) - 1/20*sqrt(5)*sqrt(2*sqrt(5) - 2
)*log(-(2*sqrt(x^4 + x^2 + 1)*(2*x^2 - sqrt(5)*x + x + 2) - (x^4 + 3*x^2 + sqrt(5)*(x^4 + x^2 + 1) + 1)*sqrt(2
*sqrt(5) - 2))/(x^4 + x^3 + x^2 + x + 1)) - arctan(x/sqrt(x^4 + x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{4} + x^{3} + x^{2} + x + 1\right )} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.56, size = 186, normalized size = 1.40

method	result	size
elliptic	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (\sqrt {x^{4}+x^{2}+1}-x^{2}\right )^{2}+\left (-15 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} \right ) \left (\sqrt {x^{4}+x^{2}+1}-x^{2}\right )-15 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +2\right )\right )}{2}+\frac {\left (-\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}-\frac {2 \sqrt {5}\, \arctanh \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\right ) \sqrt {2}}{2}\)	\(186\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \ln \left (\frac {25 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{4} x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{2}-5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right )+2 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right )-5 \sqrt {x^{4}+x^{2}+1}}{5 x \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}-x^{2}-1}\right )}{5}-\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {-25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{5} x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{3} x^{2}-15 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{3} x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{3}-3 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) x^{2}+\sqrt {x^{4}+x^{2}+1}-3 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{5 x \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+x^{2}+x +1}\right )-\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) x -\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\)	\(573\)
default	\(\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticE \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (6 \underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+11 x^{2}-6 \underline {\hspace {1.25 ex}}\alpha +4\right )}{22 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha }}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{10}\)	\(712\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sum(_R*ln(((x^4+x^2+1)^(1/2)-x^2)^2+(-15*_R^3+5*_R^2-4*_R)*((x^4+x^2+1)^(1/2)-x^2)-15*_R^3+5*_R^2-4*_R+2)
,_R=RootOf(25*_Z^4+5*_Z^2-1))+1/2*(-2/5*5^(1/2)/(5^(1/2)-1)^(1/2)*arctan((x^4+x^2+1)^(1/2)*2^(1/2)/x/(5^(1/2)-
1)^(1/2))-2/5*5^(1/2)/(5^(1/2)+1)^(1/2)*arctanh((x^4+x^2+1)^(1/2)*2^(1/2)/x/(5^(1/2)+1)^(1/2))+2^(1/2)*arctan(
1/x*(x^4+x^2+1)^(1/2)))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{4} + x^{3} + x^{2} + x + 1\right )} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^4 + x^2 + 1)*(x^2 - 1)/((x^4 + x^3 + x^2 + x + 1)*(x^2 + 1)), 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 )\,\sqrt {x^4+x^2+1}}{\left (x^2+1\right )\,\left (x^4+x^3+x^2+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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