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

Optimal. Leaf size=293 \[ \frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}-x^2}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}-x^2}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}+x^2}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^4-1}}{\sqrt {x^4-1}+x^2}\right ) \]

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Rubi [C]  time = 0.28, antiderivative size = 298, normalized size of antiderivative = 1.02, number of steps used = 28, number of rules used = 15, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {28, 1428, 416, 530, 240, 212, 206, 203, 377, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt [8]{-1} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (-1)^{5/8} \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}+1\right )}{\sqrt {2}}+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt [8]{-1} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{x^4-1}}\right )-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{x^4-1}}+\frac {x^2}{\sqrt {x^4-1}}+\sqrt [4]{-1}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{x^4-1}}-\frac {(-1)^{3/4} x^2}{\sqrt {x^4-1}}+1\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x^4 + x^8)/((-1 + x^4)^(1/4)*(1 - 2*x^4 + 2*x^8)),x]

[Out]

ArcTan[x/(-1 + x^4)^(1/4)]/4 - (1/8 + I/8)*(-1)^(1/8)*ArcTan[((-1)^(7/8)*x)/(-1 + x^4)^(1/4)] - ((1/8 + I/8)*(
-1)^(5/8)*ArcTan[1 - ((-1)^(7/8)*Sqrt[2]*x)/(-1 + x^4)^(1/4)])/Sqrt[2] + ((1/8 + I/8)*(-1)^(5/8)*ArcTan[1 + ((
-1)^(7/8)*Sqrt[2]*x)/(-1 + x^4)^(1/4)])/Sqrt[2] + ArcTanh[x/(-1 + x^4)^(1/4)]/4 - (1/8 + I/8)*(-1)^(1/8)*ArcTa
nh[((-1)^(7/8)*x)/(-1 + x^4)^(1/4)] - ((1/16 + I/16)*(-1)^(5/8)*Log[(-1)^(1/4) + x^2/Sqrt[-1 + x^4] + ((-1)^(1
/8)*Sqrt[2]*x)/(-1 + x^4)^(1/4)])/Sqrt[2] + ((1/16 + I/16)*(-1)^(5/8)*Log[1 - ((-1)^(3/4)*x^2)/Sqrt[-1 + x^4]
+ ((-1)^(7/8)*Sqrt[2]*x)/(-1 + x^4)^(1/4)])/Sqrt[2]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 203

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

Rule 204

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

Rule 206

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

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 530

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[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, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 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, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1428

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[(2*c)/r, Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[(2*c)/r, Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

Rubi steps

\begin {align*} \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx &=\int \frac {\left (-1+x^4\right )^{7/4}}{1-2 x^4+2 x^8} \, dx\\ &=-\left (2 i \int \frac {\left (-1+x^4\right )^{7/4}}{(-2-2 i)+4 x^4} \, dx\right )+2 i \int \frac {\left (-1+x^4\right )^{7/4}}{(-2+2 i)+4 x^4} \, dx\\ &=-\left (\frac {1}{8} i \int \frac {(14-2 i)-(20-8 i) x^4}{\sqrt [4]{-1+x^4} \left ((-2-2 i)+4 x^4\right )} \, dx\right )+\frac {1}{8} i \int \frac {(14+2 i)-(20+8 i) x^4}{\sqrt [4]{-1+x^4} \left ((-2+2 i)+4 x^4\right )} \, dx\\ &=-\left (\left (-\frac {1}{4}-\frac {5 i}{8}\right ) \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx\right )+\left (\frac {1}{4}-\frac {5 i}{8}\right ) \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx-\int \frac {1}{\sqrt [4]{-1+x^4} \left ((-2-2 i)+4 x^4\right )} \, dx-\int \frac {1}{\sqrt [4]{-1+x^4} \left ((-2+2 i)+4 x^4\right )} \, dx\\ &=-\left (\left (-\frac {1}{4}-\frac {5 i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )+\left (\frac {1}{4}-\frac {5 i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{(-2+2 i)-(2+2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{(-2-2 i)-(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\left (\left (-\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )-\left (-\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\left (\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\left (\frac {1}{8}-\frac {5 i}{16}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} (-1)^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}-x^2}{(-2-2 i)-(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} (-1)^{3/4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1}+x^2}{(-2-2 i)-(2-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}\\ &=\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}+2 x}{-\sqrt [4]{-1}-\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [8]{-1} \sqrt {2}-2 x}{-\sqrt [4]{-1}+\sqrt [8]{-1} \sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}\\ &=\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {-1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {1}{8} (-1)^{7/8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{8} (-1)^{7/8} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tan ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {1}{8} (-1)^{7/8} \tan ^{-1}\left (1-\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} (-1)^{7/8} \tan ^{-1}\left (1+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {(-1)^{3/8} \tanh ^{-1}\left (\frac {(-1)^{7/8} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (\sqrt [4]{-1}+\frac {x^2}{\sqrt {-1+x^4}}+\frac {\sqrt [8]{-1} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) (-1)^{5/8} \log \left (1-\frac {(-1)^{3/4} x^2}{\sqrt {-1+x^4}}+\frac {(-1)^{7/8} \sqrt {2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x^4+x^8}{\sqrt [4]{-1+x^4} \left (1-2 x^4+2 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 - 2*x^4 + x^8)/((-1 + x^4)^(1/4)*(1 - 2*x^4 + 2*x^8)),x]

[Out]

Integrate[(1 - 2*x^4 + x^8)/((-1 + x^4)^(1/4)*(1 - 2*x^4 + 2*x^8)), x]

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IntegrateAlgebraic [A]  time = 1.11, size = 273, normalized size = 0.93 \begin {gather*} \frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )+\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 2*x^4 + x^8)/((-1 + x^4)^(1/4)*(1 - 2*x^4 + 2*x^8)),x]

[Out]

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

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fricas [B]  time = 0.60, size = 1982, normalized size = 6.76

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(2)*sqrt(-sqrt(2) + 2)*arctan(-(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) - (x*(sqrt(2) + 2) - x)
*sqrt(-sqrt(2) + 2) - 2*x*sqrt((2*x^2 + (x^4 - 1)^(1/4)*(sqrt(2)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt
(2) + 2) - (sqrt(2)*x*(sqrt(2) + 2) - sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^4 - 1))/x^2) + 2*sqrt(2)*(x^4
- 1)^(1/4))/(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) + (x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2))) + 1/16
*sqrt(2)*sqrt(-sqrt(2) + 2)*arctan((x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) - (x*(sqrt(2) + 2) - x)*sqrt
(-sqrt(2) + 2) + 2*x*sqrt((2*x^2 - (x^4 - 1)^(1/4)*(sqrt(2)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt(2) +
2) - (sqrt(2)*x*(sqrt(2) + 2) - sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^4 - 1))/x^2) - 2*sqrt(2)*(x^4 - 1)^
(1/4))/(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) + (x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2))) + 1/16*sqrt
(2)*sqrt(sqrt(2) + 2)*arctan((x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) + (x*(sqrt(2) + 2) - x)*sqrt(-sqrt
(2) + 2) - 2*x*sqrt((2*x^2 + (x^4 - 1)^(1/4)*(sqrt(2)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt(2) + 2) +
(sqrt(2)*x*(sqrt(2) + 2) - sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^4 - 1))/x^2) + 2*sqrt(2)*(x^4 - 1)^(1/4))
/(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) - (x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2))) + 1/16*sqrt(2)*sq
rt(sqrt(2) + 2)*arctan(-(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) + (x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) +
2) + 2*x*sqrt((2*x^2 - (x^4 - 1)^(1/4)*(sqrt(2)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt
(2)*x*(sqrt(2) + 2) - sqrt(2)*x)*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^4 - 1))/x^2) - 2*sqrt(2)*(x^4 - 1)^(1/4))/(x*(
sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) - (x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2))) + 1/16*(sqrt(sqrt(2) +
2) + sqrt(-sqrt(2) + 2))*arctan(-((x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2) - 2*x*sqrt((x^2 + (x^4 - 1)^(1/4)*
(x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2) + sqrt(x^4 - 1))/x^2) + 2*(x^4 - 1)^(1/4))/(x*(sqrt(2) + 2)^(3/2) - 3
*x*sqrt(sqrt(2) + 2))) + 1/16*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*arctan(((x*(sqrt(2) + 2) - x)*sqrt(-sqr
t(2) + 2) + 2*x*sqrt((x^2 - (x^4 - 1)^(1/4)*(x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2) + sqrt(x^4 - 1))/x^2) - 2
*(x^4 - 1)^(1/4))/(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2))) + 1/16*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) +
2))*arctan(-(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2) - 2*x*sqrt((x^2 + (x^4 - 1)^(1/4)*(x*(sqrt(2) + 2)
^(3/2) - 3*x*sqrt(sqrt(2) + 2)) + sqrt(x^4 - 1))/x^2) + 2*(x^4 - 1)^(1/4))/((x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2
) + 2))) + 1/16*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*arctan((x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2)
+ 2*x*sqrt((x^2 - (x^4 - 1)^(1/4)*(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2)) + sqrt(x^4 - 1))/x^2) - 2*(
x^4 - 1)^(1/4))/((x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2))) + 1/64*sqrt(2)*sqrt(-sqrt(2) + 2)*log(32*(2*x^2 +
(x^4 - 1)^(1/4)*(sqrt(2)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2)*x*(sqrt(2) + 2) - sq
rt(2)*x)*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^4 - 1))/x^2) - 1/64*sqrt(2)*sqrt(-sqrt(2) + 2)*log(32*(2*x^2 - (x^4 -
1)^(1/4)*(sqrt(2)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt(2) + 2) + (sqrt(2)*x*(sqrt(2) + 2) - sqrt(2)*x
)*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^4 - 1))/x^2) - 1/64*sqrt(2)*sqrt(sqrt(2) + 2)*log(32*(2*x^2 + (x^4 - 1)^(1/4)
*(sqrt(2)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2)*x*(sqrt(2) + 2) - sqrt(2)*x)*sqrt(-
sqrt(2) + 2)) + 2*sqrt(x^4 - 1))/x^2) + 1/64*sqrt(2)*sqrt(sqrt(2) + 2)*log(32*(2*x^2 - (x^4 - 1)^(1/4)*(sqrt(2
)*x*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*x*sqrt(sqrt(2) + 2) - (sqrt(2)*x*(sqrt(2) + 2) - sqrt(2)*x)*sqrt(-sqrt(2)
+ 2)) + 2*sqrt(x^4 - 1))/x^2) + 1/64*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*log(256*(x^2 + (x^4 - 1)^(1/4)*(
x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2) + sqrt(x^4 - 1))/x^2) - 1/64*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*
log(256*(x^2 - (x^4 - 1)^(1/4)*(x*(sqrt(2) + 2) - x)*sqrt(-sqrt(2) + 2) + sqrt(x^4 - 1))/x^2) + 1/64*(sqrt(sqr
t(2) + 2) + sqrt(-sqrt(2) + 2))*log(256*(x^2 + (x^4 - 1)^(1/4)*(x*(sqrt(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2))
+ sqrt(x^4 - 1))/x^2) - 1/64*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*log(256*(x^2 - (x^4 - 1)^(1/4)*(x*(sqrt
(2) + 2)^(3/2) - 3*x*sqrt(sqrt(2) + 2)) + sqrt(x^4 - 1))/x^2) - 1/4*arctan((x^4 - 1)^(1/4)/x) + 1/8*log((x + (
x^4 - 1)^(1/4))/x) - 1/8*log(-(x - (x^4 - 1)^(1/4))/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 13.14, size = 1006, normalized size = 3.43

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{8}+\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+2 x^{4}-1\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (-\frac {\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{2}-\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}-\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+x^{4}-1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{2}+\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}-\left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}-2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+x^{4}-1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+4 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}+4 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{2}+8 \left (x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}-\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-x^{4}+1}\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} x^{4}-4 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}-4 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{2}+8 \left (x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}-\RootOf \left (\textit {\_Z}^{4}+4 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )^{3}}{\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-x^{4}+1}\right )}{32}\)	\(1006\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*RootOf(_Z^2+1)*ln(-2*RootOf(_Z^2+1)*(x^4-1)^(1/2)*x^2+2*RootOf(_Z^2+1)*x^4+2*(x^4-1)^(3/4)*x-2*x^3*(x^4-1)
^(1/4)-RootOf(_Z^2+1))+1/8*ln(2*(x^4-1)^(3/4)*x+2*x^2*(x^4-1)^(1/2)+2*x^3*(x^4-1)^(1/4)+2*x^4-1)-1/16*RootOf(_
Z^4+4*RootOf(_Z^2+1))*ln(-((x^4-1)^(1/2)*RootOf(_Z^4+4*RootOf(_Z^2+1))^3*x^2-(x^4-1)^(1/4)*RootOf(_Z^2+1)*Root
Of(_Z^4+4*RootOf(_Z^2+1))^2*x^3-(x^4-1)^(1/4)*RootOf(_Z^4+4*RootOf(_Z^2+1))^2*x^3+RootOf(_Z^4+4*RootOf(_Z^2+1)
)*RootOf(_Z^2+1)*x^4+2*RootOf(_Z^2+1)*(x^4-1)^(3/4)*x-RootOf(_Z^4+4*RootOf(_Z^2+1))*x^4+2*(x^4-1)^(3/4)*x+Root
Of(_Z^4+4*RootOf(_Z^2+1)))/(RootOf(_Z^2+1)*x^4+x^4-1))+1/16*RootOf(_Z^2+1)*RootOf(_Z^4+4*RootOf(_Z^2+1))*ln((-
(x^4-1)^(1/2)*RootOf(_Z^4+4*RootOf(_Z^2+1))^3*x^2+(x^4-1)^(1/4)*RootOf(_Z^2+1)*RootOf(_Z^4+4*RootOf(_Z^2+1))^2
*x^3-(x^4-1)^(1/4)*RootOf(_Z^4+4*RootOf(_Z^2+1))^2*x^3+RootOf(_Z^4+4*RootOf(_Z^2+1))*RootOf(_Z^2+1)*x^4+2*Root
Of(_Z^2+1)*(x^4-1)^(3/4)*x-RootOf(_Z^4+4*RootOf(_Z^2+1))*x^4-2*(x^4-1)^(3/4)*x+RootOf(_Z^4+4*RootOf(_Z^2+1)))/
(RootOf(_Z^2+1)*x^4+x^4-1))+1/32*RootOf(_Z^4+4*RootOf(_Z^2+1))^3*ln(-(RootOf(_Z^4+4*RootOf(_Z^2+1))^3*RootOf(_
Z^2+1)^2*x^4+2*RootOf(_Z^4+4*RootOf(_Z^2+1))^3*RootOf(_Z^2+1)*x^4+4*(x^4-1)^(1/4)*RootOf(_Z^2+1)*RootOf(_Z^4+4
*RootOf(_Z^2+1))^2*x^3+RootOf(_Z^4+4*RootOf(_Z^2+1))^3*x^4+4*RootOf(_Z^4+4*RootOf(_Z^2+1))*RootOf(_Z^2+1)*(x^4
-1)^(1/2)*x^2+4*(x^4-1)^(1/2)*RootOf(_Z^4+4*RootOf(_Z^2+1))*x^2+8*(x^4-1)^(3/4)*x-RootOf(_Z^2+1)*RootOf(_Z^4+4
*RootOf(_Z^2+1))^3-RootOf(_Z^4+4*RootOf(_Z^2+1))^3)/(RootOf(_Z^2+1)*x^4-x^4+1))-1/32*RootOf(_Z^2+1)*RootOf(_Z^
4+4*RootOf(_Z^2+1))^3*ln(-(-RootOf(_Z^4+4*RootOf(_Z^2+1))^3*RootOf(_Z^2+1)^2*x^4-4*(x^4-1)^(1/4)*RootOf(_Z^2+1
)*RootOf(_Z^4+4*RootOf(_Z^2+1))^2*x^3+RootOf(_Z^4+4*RootOf(_Z^2+1))^3*x^4+4*RootOf(_Z^4+4*RootOf(_Z^2+1))*Root
Of(_Z^2+1)*(x^4-1)^(1/2)*x^2-4*(x^4-1)^(1/2)*RootOf(_Z^4+4*RootOf(_Z^2+1))*x^2+8*(x^4-1)^(3/4)*x+RootOf(_Z^2+1
)*RootOf(_Z^4+4*RootOf(_Z^2+1))^3-RootOf(_Z^4+4*RootOf(_Z^2+1))^3)/(RootOf(_Z^2+1)*x^4-x^4+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 2 \, x^{4} + 1}{{\left (2 \, x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^8-2\,x^4+1}{{\left (x^4-1\right )}^{1/4}\,\left (2\,x^8-2\,x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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