\(\int \frac {x^2}{1-x^4+x^8} \, dx\) [361]

Optimal result

Integrand size = 16, antiderivative size = 355 \[ \int \frac {x^2}{1-x^4+x^8} \, dx=\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \]

[Out]

1/4*arctan((-2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*2^(1/2)-1/6*6^(1/2))-1/4*arctan((2*x
+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*2^(1/2)-1/6*6^(1/2))+1/8*ln(1+x^2-x*(1/2*6^(1/2)-1/2
*2^(1/2)))/(3/2*2^(1/2)-1/2*6^(1/2))-1/8*ln(1+x^2+x*(1/2*6^(1/2)-1/2*2^(1/2)))/(3/2*2^(1/2)-1/2*6^(1/2))-1/4*a
rctan((-2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)+1/6*6^(1/2))+1/4*arctan((2*x+1/2*
6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)+1/6*6^(1/2))-1/8*ln(1+x^2-x*(1/2*6^(1/2)+1/2*2^(1
/2)))/(3/2*2^(1/2)+1/2*6^(1/2))+1/8*ln(1+x^2+x*(1/2*6^(1/2)+1/2*2^(1/2)))/(3/2*2^(1/2)+1/2*6^(1/2))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1387, 1108, 648, 632, 210, 642} \[ \int \frac {x^2}{1-x^4+x^8} \, dx=\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )+\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \]

[In]

Int[x^2/(1 - x^4 + x^8),x]

[Out]

(Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]])/4 - (Sqrt[(2 + Sqrt[3])/3]*ArcTan[
(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/4 - (Sqrt[(2 - Sqrt[3])/3]*ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt
[2 + Sqrt[3]]])/4 + (Sqrt[(2 + Sqrt[3])/3]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/4 + Log[1 - Sq
rt[2 - Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 - Sqrt[3])]) - Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 - Sqrt[3]
)]) - Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2]/(8*Sqrt[3*(2 + Sqrt[3])]) + Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2]/(8*Sqr
t[3*(2 + Sqrt[3])])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1108

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 1387

Int[(x_)^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[
2*q - b/c, 2]}, Dist[1/(2*c*r), Int[x^(m - n/2)/(q - r*x^(n/2) + x^n), x], x] - Dist[1/(2*c*r), Int[x^(m - n/2
)/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0
] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, 3*(n/2)] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {1}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}} \\ & = -\frac {\int \frac {\sqrt {2-\sqrt {3}}-x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {2-\sqrt {3}}+x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}-x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}+x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}} \\ & = -\frac {\int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}-\frac {\int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \\ & = \frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )}{4 \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )}{4 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{1-x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \]

[In]

Integrate[x^2/(1 - x^4 + x^8),x]

[Out]

RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1 + 2*#1^5) & ]/4

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.11

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\)	\(40\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\)	\(40\)

[In]

int(x^2/(x^8-x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/4*sum(_R^2/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.54 \[ \int \frac {x^2}{1-x^4+x^8} \, dx=-\frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (i \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \sqrt {i \, \sqrt {3} + 1} + 24 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (-i \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \sqrt {i \, \sqrt {3} + 1} + 24 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (i \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \sqrt {i \, \sqrt {3} + 1} + 24 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (-i \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \sqrt {i \, \sqrt {3} + 1} + 24 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (i \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \sqrt {-i \, \sqrt {3} + 1} + 24 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (-i \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \sqrt {-i \, \sqrt {3} + 1} + 24 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (i \, \sqrt {3} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \sqrt {-i \, \sqrt {3} + 1} + 24 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} {\left (-i \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \sqrt {-i \, \sqrt {3} + 1} + 24 \, x\right ) \]

[In]

integrate(x^2/(x^8-x^4+1),x, algorithm="fricas")

[Out]

-1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(I*sqrt(3) + 1))*log(sqrt(6)*(I*sqrt(3)*sqrt(2) + 3*sqrt(2))*sqrt(sqrt(2)*sqrt(
I*sqrt(3) + 1))*sqrt(I*sqrt(3) + 1) + 24*x) + 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(I*sqrt(3) + 1))*log(sqrt(6)*(-I*s
qrt(3)*sqrt(2) - 3*sqrt(2))*sqrt(sqrt(2)*sqrt(I*sqrt(3) + 1))*sqrt(I*sqrt(3) + 1) + 24*x) + 1/24*sqrt(6)*sqrt(
-sqrt(2)*sqrt(I*sqrt(3) + 1))*log(sqrt(6)*(I*sqrt(3)*sqrt(2) + 3*sqrt(2))*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 1))*s
qrt(I*sqrt(3) + 1) + 24*x) - 1/24*sqrt(6)*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 1))*log(sqrt(6)*(-I*sqrt(3)*sqrt(2) -
3*sqrt(2))*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 1))*sqrt(I*sqrt(3) + 1) + 24*x) + 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(-I
*sqrt(3) + 1))*log(sqrt(6)*(I*sqrt(3)*sqrt(2) - 3*sqrt(2))*sqrt(sqrt(2)*sqrt(-I*sqrt(3) + 1))*sqrt(-I*sqrt(3)
+ 1) + 24*x) - 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(-I*sqrt(3) + 1))*log(sqrt(6)*(-I*sqrt(3)*sqrt(2) + 3*sqrt(2))*sq
rt(sqrt(2)*sqrt(-I*sqrt(3) + 1))*sqrt(-I*sqrt(3) + 1) + 24*x) - 1/24*sqrt(6)*sqrt(-sqrt(2)*sqrt(-I*sqrt(3) + 1
))*log(sqrt(6)*(I*sqrt(3)*sqrt(2) - 3*sqrt(2))*sqrt(-sqrt(2)*sqrt(-I*sqrt(3) + 1))*sqrt(-I*sqrt(3) + 1) + 24*x
) + 1/24*sqrt(6)*sqrt(-sqrt(2)*sqrt(-I*sqrt(3) + 1))*log(sqrt(6)*(-I*sqrt(3)*sqrt(2) + 3*sqrt(2))*sqrt(-sqrt(2
)*sqrt(-I*sqrt(3) + 1))*sqrt(-I*sqrt(3) + 1) + 24*x)

Sympy [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.07 \[ \int \frac {x^2}{1-x^4+x^8} \, dx=\operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 442368 t^{7} - 192 t^{3} + x \right )} \right )\right )} \]

[In]

integrate(x**2/(x**8-x**4+1),x)

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-442368*_t**7 - 192*_t**3 + x)))

Maxima [F]

\[ \int \frac {x^2}{1-x^4+x^8} \, dx=\int { \frac {x^{2}}{x^{8} - x^{4} + 1} \,d x } \]

[In]

integrate(x^2/(x^8-x^4+1),x, algorithm="maxima")

[Out]

integrate(x^2/(x^8 - x^4 + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{1-x^4+x^8} \, dx=\frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]

[In]

integrate(x^2/(x^8-x^4+1),x, algorithm="giac")

[Out]

1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) + 1/24*(sqrt(6) - 3*sqrt(2))*
arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) + sqr
t(2))/(sqrt(6) - sqrt(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2)))
- 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 - 1
/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) + 1/48*(sq
rt(6) + 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{1-x^4+x^8} \, dx=-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \]

[In]

int(x^2/(x^8 - x^4 + 1),x)

[Out]

(3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*(3^(1/2)*1i + 1)) - (3^(1/2)*x*(8 - 3^(1/2)*8i)^(1/4))/(2*(3^(1
/2)*1i + 1)))*(8 - 3^(1/2)*8i)^(1/4))/12 - (3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4))/(2*(3^(1/2)*1i + 1)) + (3^
(1/2)*x*(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*(3^(1/2)*1i + 1)))*(8 - 3^(1/2)*8i)^(1/4)*1i)/12 - (2^(3/4)*3^(1/2)*atan
((2^(3/4)*x*(3^(1/2)*1i + 1)^(1/4))/(2*(3^(1/2)*1i - 1)) - (2^(3/4)*3^(1/2)*x*(3^(1/2)*1i + 1)^(1/4)*1i)/(2*(3
^(1/2)*1i - 1)))*(3^(1/2)*1i + 1)^(1/4)*1i)/12 + (2^(3/4)*3^(1/2)*atan((2^(3/4)*x*(3^(1/2)*1i + 1)^(1/4)*1i)/(
2*(3^(1/2)*1i - 1)) + (2^(3/4)*3^(1/2)*x*(3^(1/2)*1i + 1)^(1/4))/(2*(3^(1/2)*1i - 1)))*(3^(1/2)*1i + 1)^(1/4))
/12