\(\int \frac {x^4}{1-x^4+x^8} \, dx\) [360]

Optimal result

Integrand size = 16, antiderivative size = 347 \[ \int \frac {x^4}{1-x^4+x^8} \, dx=\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \]

[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)))/(3/2*2^(1/2)-1/2*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)))/(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))+1/8*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*
arctan((-2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))/(3/2*2^(1/2)+1/2*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)))/(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))-1/8*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 = 347, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1387, 1141, 1175, 632, 210, 1178, 642} \[ \int \frac {x^4}{1-x^4+x^8} \, dx=\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}} \]

[In]

Int[x^4/(1 - x^4 + x^8),x]

[Out]

ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) - ArcTan[(Sqrt[2 + Sqrt[3]] - 2*
x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) - ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[
3*(2 + Sqrt[3])]) + ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) - 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 1141

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

Rule 1175

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 1178

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

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 {x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}} \\ & = -\frac {\int \frac {1-x^2}{1-\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}+\frac {\int \frac {1+x^2}{1-\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}+\frac {\int \frac {1-x^2}{1+\sqrt {3} x^2+x^4} \, dx}{4 \sqrt {3}}-\frac {\int \frac {1+x^2}{1+\sqrt {3} x^2+x^4} \, dx}{4 \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 {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 = 39, normalized size of antiderivative = 0.11 \[ \int \frac {x^4}{1-x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ] \]

[In]

Integrate[x^4/(1 - x^4 + x^8),x]

[Out]

RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1)/(-1 + 2*#1^4) & ]/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 = 0.12

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

[In]

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

[Out]

1/4*sum(_R^4/(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 = 0.26 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.13 \[ \int \frac {x^4}{1-x^4+x^8} \, dx=-\frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (i \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (-i \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} + 6 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (i \, \sqrt {6} \sqrt {3} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (-i \, \sqrt {6} \sqrt {3} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (i \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} + 6 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (-i \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} + 6 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (i \, \sqrt {6} \sqrt {3} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} + 6 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (-i \, \sqrt {6} \sqrt {3} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} + 6 \, x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.07 \[ \int \frac {x^4}{1-x^4+x^8} \, dx=\operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 18432 t^{5} + 4 t + x \right )} \right )\right )} \]

[In]

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

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-18432*_t**5 + 4*_t + x)))

Maxima [F]

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

[In]

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

[Out]

integrate(x^4/(x^8 - x^4 + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.73 \[ \int \frac {x^4}{1-x^4+x^8} \, dx=\frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]

[In]

integrate(x^4/(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 = 8.25 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.37 \[ \int \frac {x^4}{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 (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\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 (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {3}\,\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {8-\sqrt {3}\,8{}\mathrm {i}}}{4}\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 (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}-\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\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 (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}+\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (\frac {\sqrt {2}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}}{2}-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \]

[In]

int(x^4/(x^8 - x^4 + 1),x)

[Out]

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