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

Optimal. Leaf size=105 \[ \frac {1}{8} \text {RootSum}\left [2 \text {$\#$1}^8-6 \text {$\#$1}^4+5\& ,\frac {-3 \text {$\#$1}^4 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )+3 \text {$\#$1}^4 \log (x)+5 \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )-5 \log (x)}{2 \text {$\#$1}^7-3 \text {$\#$1}^3}\& \right ]-\frac {\sqrt [4]{x^4-1}}{x} \]

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Rubi [C]  time = 1.53, antiderivative size = 295, normalized size of antiderivative = 2.81, number of steps used = 42, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {6728, 277, 331, 298, 203, 206, 1528, 511, 510, 1518, 494} \begin {gather*} -\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \sqrt [4]{x^4-1} x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\left (-\frac {1}{2}+\frac {i}{2}\right ) x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [4]{x^4-1} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (-\frac {1}{2}-\frac {i}{2}\right ) x^4,x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\sqrt [4]{x^4-1}}{x}-\left (\frac {1}{4}+\frac {i}{2}\right ) \left (\frac {3}{5}-\frac {i}{5}\right )^{3/4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{x^4-1}}\right )-\left (\frac {1}{4}-\frac {i}{2}\right ) \left (\frac {3}{5}+\frac {i}{5}\right )^{3/4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{x^4-1}}\right )+\left (\frac {1}{4}+\frac {i}{2}\right ) \left (\frac {3}{5}-\frac {i}{5}\right )^{3/4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{x^4-1}}\right )+\left (\frac {1}{4}-\frac {i}{2}\right ) \left (\frac {3}{5}+\frac {i}{5}\right )^{3/4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{x^4-1}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((-1 + x^4)^(1/4)/x) - ((1/12 + I/12)*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, (-1/2 + I/2)*x^4]
)/(1 - x^4)^(1/4) - ((1/12 - I/12)*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (-1/2 - I/2)*x^4, x^4])/(1
- x^4)^(1/4) - (1/4 + I/2)*(3/5 - I/5)^(3/4)*ArcTan[x/((3/5 - I/5)^(1/4)*(-1 + x^4)^(1/4))] - (1/4 - I/2)*(3/
5 + I/5)^(3/4)*ArcTan[x/((3/5 + I/5)^(1/4)*(-1 + x^4)^(1/4))] + (1/4 + I/2)*(3/5 - I/5)^(3/4)*ArcTanh[x/((3/5
- I/5)^(1/4)*(-1 + x^4)^(1/4))] + (1/4 - I/2)*(3/5 + I/5)^(3/4)*ArcTanh[x/((3/5 + I/5)^(1/4)*(-1 + x^4)^(1/4))
]

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 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 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

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

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]

Rule 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1518

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol
] :> Dist[(e*f^n)/c, Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[((f*x)^(m - n)*(d + e*x^n
)^(q - 1)*Simp[a*e - (c*d - b*e)*x^n, x])/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && E
qQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n
- 1]

Rule 1528

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx &=\int \left (\frac {\sqrt [4]{-1+x^4}}{x^2}+\frac {x^2 \left (-1-x^4\right ) \sqrt [4]{-1+x^4}}{2+2 x^4+x^8}\right ) \, dx\\ &=\int \frac {\sqrt [4]{-1+x^4}}{x^2} \, dx+\int \frac {x^2 \left (-1-x^4\right ) \sqrt [4]{-1+x^4}}{2+2 x^4+x^8} \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}+\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx+\int \left (-\frac {x^2 \sqrt [4]{-1+x^4}}{2+2 x^4+x^8}-\frac {x^6 \sqrt [4]{-1+x^4}}{2+2 x^4+x^8}\right ) \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\int \frac {x^2 \sqrt [4]{-1+x^4}}{2+2 x^4+x^8} \, dx-\int \frac {x^6 \sqrt [4]{-1+x^4}}{2+2 x^4+x^8} \, dx+\operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx+\int \frac {x^2 \left (2+3 x^4\right )}{\left (-1+x^4\right )^{3/4} \left (2+2 x^4+x^8\right )} \, dx-\int \left (\frac {i x^2 \sqrt [4]{-1+x^4}}{(-2+2 i)-2 x^4}+\frac {i x^2 \sqrt [4]{-1+x^4}}{(2+2 i)+2 x^4}\right ) \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-i \int \frac {x^2 \sqrt [4]{-1+x^4}}{(-2+2 i)-2 x^4} \, dx-i \int \frac {x^2 \sqrt [4]{-1+x^4}}{(2+2 i)+2 x^4} \, dx+\int \left (\frac {2 x^2}{\left (-1+x^4\right )^{3/4} \left (2+2 x^4+x^8\right )}+\frac {3 x^6}{\left (-1+x^4\right )^{3/4} \left (2+2 x^4+x^8\right )}\right ) \, dx-\operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+2 \int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (2+2 x^4+x^8\right )} \, dx+3 \int \frac {x^6}{\left (-1+x^4\right )^{3/4} \left (2+2 x^4+x^8\right )} \, dx-\frac {\left (i \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{(-2+2 i)-2 x^4} \, dx}{\sqrt [4]{1-x^4}}-\frac {\left (i \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{(2+2 i)+2 x^4} \, dx}{\sqrt [4]{1-x^4}}\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\left (-\frac {1}{2}+\frac {i}{2}\right ) x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (-\frac {1}{2}-\frac {i}{2}\right ) x^4,x^4\right )}{\sqrt [4]{1-x^4}}+2 \int \left (\frac {i x^2}{\left ((-2+2 i)-2 x^4\right ) \left (-1+x^4\right )^{3/4}}+\frac {i x^2}{\left (-1+x^4\right )^{3/4} \left ((2+2 i)+2 x^4\right )}\right ) \, dx+3 \int \left (-\frac {(1+i) x^2}{\left ((-2+2 i)-2 x^4\right ) \left (-1+x^4\right )^{3/4}}+\frac {(1-i) x^2}{\left (-1+x^4\right )^{3/4} \left ((2+2 i)+2 x^4\right )}\right ) \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\left (-\frac {1}{2}+\frac {i}{2}\right ) x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (-\frac {1}{2}-\frac {i}{2}\right ) x^4,x^4\right )}{\sqrt [4]{1-x^4}}+(-3-3 i) \int \frac {x^2}{\left ((-2+2 i)-2 x^4\right ) \left (-1+x^4\right )^{3/4}} \, dx+2 i \int \frac {x^2}{\left ((-2+2 i)-2 x^4\right ) \left (-1+x^4\right )^{3/4}} \, dx+2 i \int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left ((2+2 i)+2 x^4\right )} \, dx+(3-3 i) \int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left ((2+2 i)+2 x^4\right )} \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\left (-\frac {1}{2}+\frac {i}{2}\right ) x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (-\frac {1}{2}-\frac {i}{2}\right ) x^4,x^4\right )}{\sqrt [4]{1-x^4}}+(-3-3 i) \operatorname {Subst}\left (\int \frac {x^2}{(-2+2 i)+(4-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+2 i \operatorname {Subst}\left (\int \frac {x^2}{(2+2 i)-(4+2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+2 i \operatorname {Subst}\left (\int \frac {x^2}{(-2+2 i)+(4-2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+(3-3 i) \operatorname {Subst}\left (\int \frac {x^2}{(2+2 i)-(4+2 i) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\left (-\frac {1}{2}+\frac {i}{2}\right ) x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (-\frac {1}{2}-\frac {i}{2}\right ) x^4,x^4\right )}{\sqrt [4]{1-x^4}}+-\frac {\left (\frac {3}{4}+\frac {9 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-i}+\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+-\frac {\left (\frac {3}{4}-\frac {9 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+i}+\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+-\frac {\left (\frac {1}{2}+i\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+i}+\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+-\frac {\left (\frac {1}{2}-i\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-i}+\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+\frac {\left (\frac {1}{2}-i\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-i}-\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+\frac {\left (\frac {1}{2}+i\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+i}-\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+\frac {\left (\frac {3}{4}-\frac {9 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+i}-\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}+\frac {\left (\frac {3}{4}+\frac {9 i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-i}-\sqrt {5} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {5}}\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\left (-\frac {1}{2}+\frac {i}{2}\right ) x^4\right )}{\sqrt [4]{1-x^4}}-\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (-\frac {1}{2}-\frac {i}{2}\right ) x^4,x^4\right )}{\sqrt [4]{1-x^4}}-\frac {3}{4} i \left (\frac {3}{5}-\frac {i}{5}\right )^{3/4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )-\frac {\left (\frac {1}{2}-i\right ) \tan ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{3-i} 5^{3/4}}+\frac {3}{4} i \left (\frac {3}{5}+\frac {i}{5}\right )^{3/4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )-\frac {\left (\frac {1}{2}+i\right ) \tan ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{3+i} 5^{3/4}}+\frac {3}{4} i \left (\frac {3}{5}-\frac {i}{5}\right )^{3/4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )+\frac {\left (\frac {1}{2}-i\right ) \tanh ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}-\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{3-i} 5^{3/4}}-\frac {3}{4} i \left (\frac {3}{5}+\frac {i}{5}\right )^{3/4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )+\frac {\left (\frac {1}{2}+i\right ) \tanh ^{-1}\left (\frac {x}{\sqrt [4]{\frac {3}{5}+\frac {i}{5}} \sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{3+i} 5^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 6.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{-1+x^4} \left (2+x^4\right )}{x^2 \left (2+2 x^4+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.23, size = 105, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{-1+x^4}}{x}+\frac {1}{8} \text {RootSum}\left [5-6 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-5 \log (x)+5 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+3 \log (x) \text {$\#$1}^4-3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1 + x^4)^(1/4)/x) + RootSum[5 - 6*#1^4 + 2*#1^8 & , (-5*Log[x] + 5*Log[(-1 + x^4)^(1/4) - x*#1] + 3*Log[x]
*#1^4 - 3*Log[(-1 + x^4)^(1/4) - x*#1]*#1^4)/(-3*#1^3 + 2*#1^7) & ]/8

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

[Out]

Timed out

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giac [B]  time = 0.35, size = 396, normalized size = 3.77

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4 - 1)^(1/4)/x + 1/32768*(8*I + 8)^(33/4)*(I + 2)^(1/4)*log(I*(37572295510484393868859378145757959465275804
876800000*I + 26011589199566118832287261793217048860575557222400000)^(1/4) - (12369505812480*I - 4123168604160
)*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^17 - 1/32768*(8*I + 8)^(33/4)*(I + 2)^(1/4)*lo
g(I*(37572295510484393868859378145757959465275804876800000*I + 26011589199566118832287261793217048860575557222
400000)^(1/4) + (12369505812480*I - 4123168604160)*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2
))^17 + 1/32768*(8*I + 8)^(33/4)*(2*I + 1)^(1/4)*log(-I*(-3757229551048439386885937814575795946527580487680000
0*I + 26011589199566118832287261793217048860575557222400000)^(1/4) + (12369505812480*I + 4123168604160)*(x^4 -
1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^17 - 1/32768*(8*I + 8)^(33/4)*(2*I + 1)^(1/4)*log(-I*(
-37572295510484393868859378145757959465275804876800000*I + 260115891995661188322872617932170488605755572224000
00)^(1/4) - (12369505812480*I + 4123168604160)*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^1
7 + 1/134217728*I*(8*I + 8)^(65/4)*(I + 2)^(1/4)*log(-I*(49942147058439763468705893258915171110704453162155832
907181603209352202449245228236800000*I + 345753325789198362475656184100181953843338521891848073972795714526284
47849477465702400000)^(1/4) + (4427218577690292387840*I + 13281655733070877163520)*(x^4 - 1)^(1/4)/x)/(sqrt(sq
rt(2) + 2) + I*sqrt(-sqrt(2) + 2))^33 - 1/134217728*I*(8*I + 8)^(65/4)*(I + 2)^(1/4)*log(-I*(49942147058439763
468705893258915171110704453162155832907181603209352202449245228236800000*I + 345753325789198362475656184100181
95384333852189184807397279571452628447849477465702400000)^(1/4) - (4427218577690292387840*I + 1328165573307087
7163520)*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^33 - 1/134217728*I*(8*I + 8)^(65/4)*(2*
I + 1)^(1/4)*log(I*(-49942147058439763468705893258915171110704453162155832907181603209352202449245228236800000
*I + 34575332578919836247565618410018195384333852189184807397279571452628447849477465702400000)^(1/4) - (44272
18577690292387840*I - 13281655733070877163520)*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^3
3 + 1/134217728*I*(8*I + 8)^(65/4)*(2*I + 1)^(1/4)*log(I*(-499421470584397634687058932589151711107044531621558
32907181603209352202449245228236800000*I + 3457533257891983624756561841001819538433385218918480739727957145262
8447849477465702400000)^(1/4) + (4427218577690292387840*I - 13281655733070877163520)*(x^4 - 1)^(1/4)/x)/(sqrt(
sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^33

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maple [B]  time = 21.29, size = 4180, normalized size = 39.81 \[\text {output too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^4-1)^(1/4)/x+(-1024*ln(-(4096*x^12*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_
Z^8-24576*_Z^4+5)^4-3)^2+16384*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+409
6*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x^9-8192*x^8*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096
*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2-4*x^12*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2-3
2768*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-245
76*_Z^4+5)^4-3)^3*x^5-6*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*
x^9-8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-24576*(x^12-3*x^8+3*x^4-1)^(3/4)*
RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)*x^3+4096*x^4*Roo
tOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2+9*x^8*RootOf(_Z^4
+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2+16384*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z
^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x+12*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4
+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x^5+8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*(x^12-3*x^8+3*x^
4-1)^(1/2)*x^2+6*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+8*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^4+4096*RootOf(33554432*
_Z^8-24576*_Z^4+5)^4-3)*x^3-6*x^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2-6*(x^12-3*x^8+3*x
^4-1)^(1/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x-6*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+Root
Of(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2)/(4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^4-2*x^4-1
)/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x-1)^2/(-1+x)^2/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-
3*x+1)^2/(1+x)^2)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3
)+3/8*ln(-(4096*x^12*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^
4-3)^2+16384*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*
_Z^8-24576*_Z^4+5)^4-3)^3*x^9-8192*x^8*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_
Z^8-24576*_Z^4+5)^4-3)^2-4*x^12*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2-32768*(x^12-3*x^8+3
*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*
x^5-6*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x^9-8192*RootOf(33
554432*_Z^8-24576*_Z^4+5)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-24576*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(33554432*_Z
^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)*x^3+4096*x^4*RootOf(33554432*_Z^8-
24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2+9*x^8*RootOf(_Z^4+4096*RootOf(33554
432*_Z^8-24576*_Z^4+5)^4-3)^2+16384*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^
4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x+12*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+4096*RootOf(33554
432*_Z^8-24576*_Z^4+5)^4-3)^3*x^5+8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+6*(
x^12-3*x^8+3*x^4-1)^(1/2)*x^6+8*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)
^4-3)*x^3-6*x^4*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2-6*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf
(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x-6*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+RootOf(_Z^4+4096*RootO
f(33554432*_Z^8-24576*_Z^4+5)^4-3)^2)/(4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^4-2*x^4-1)/(8192*RootOf(335
54432*_Z^8-24576*_Z^4+5)^4*x-3*x-1)^2/(-1+x)^2/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x+1)^2/(1+x)^2)*
RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)+8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*ln(-(-16384
*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^12-524288*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5
)^6*x^9+32768*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^8-4*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^12+1048576*(x^12
-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^6*x^5+192*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*
_Z^8-24576*_Z^4+5)^2*x^9+512*RootOf(33554432*_Z^8-24576*_Z^4+5)^3*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+12288*RootOf(
33554432*_Z^8-24576*_Z^4+5)^4*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-16384*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^4+12
*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^8-524288*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^6
*x-384*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^2*x^5-512*RootOf(33554432*_Z^8-24576*_Z^4
+5)^3*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-5*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-12*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^
4+192*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^2*x+4*RootOf(33554432*_Z^8-24576*_Z^4+5))/
(4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^4-x^4+1)/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x-1)^2/(-
1+x)^2/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x+1)^2/(1+x)^2)-3*RootOf(33554432*_Z^8-24576*_Z^4+5)*ln(
-(-16384*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^12-524288*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-2457
6*_Z^4+5)^6*x^9+32768*RootOf(33554432*_Z^8-24576*_Z^4+5)^5*x^8-4*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^12+10485
76*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^6*x^5+192*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(3
3554432*_Z^8-24576*_Z^4+5)^2*x^9+512*RootOf(33554432*_Z^8-24576*_Z^4+5)^3*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+12288
*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-16384*RootOf(33554432*_Z^8-24576*_Z^4+5)^
5*x^4+12*RootOf(33554432*_Z^8-24576*_Z^4+5)*x^8-524288*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_
Z^4+5)^6*x-384*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^2*x^5-512*RootOf(33554432*_Z^8-24
576*_Z^4+5)^3*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-5*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-12*RootOf(33554432*_Z^8-24576*_Z
^4+5)*x^4+192*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^2*x+4*RootOf(33554432*_Z^8-24576*_
Z^4+5))/(4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^4-x^4+1)/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x
-1)^2/(-1+x)^2/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x+1)^2/(1+x)^2)+1/8*RootOf(_Z^4+4096*RootOf(3355
4432*_Z^8-24576*_Z^4+5)^4-3)*ln(-(-4096*x^12*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(3355
4432*_Z^8-24576*_Z^4+5)^4-3)^2+8192*x^8*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(_Z^4+4096*RootOf(33554432*
_Z^8-24576*_Z^4+5)^4-3)^2+4*x^12*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2+2*(x^12-3*x^8+3*x^
4-1)^(1/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x^9-8192*RootOf(33554432*_Z^8-24576*_Z^4
+5)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-8192*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*Root
Of(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)*x^3-4096*x^4*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*RootOf(
_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2-9*x^8*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^
4-3)^2-4*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^3*x^5+8192*RootOf
(33554432*_Z^8-24576*_Z^4+5)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+6*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+6*(x^12-3*x^8+3
*x^4-1)^(3/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)*x^3+6*x^4*RootOf(_Z^4+4096*RootOf(33554
432*_Z^8-24576*_Z^4+5)^4-3)^2+2*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)
^4-3)^3*x-6*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-RootOf(_Z^4+4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4-3)^2)/(4096*R
ootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^4-2*x^4-1)/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x-1)^2/(-1+x)^
2/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x+1)^2/(1+x)^2)+RootOf(33554432*_Z^8-24576*_Z^4+5)*ln(-(4096*
RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^12-8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^8+x^12+16*RootOf(3355443
2*_Z^8-24576*_Z^4+5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+128*RootOf(33554432*_Z^8-24576*_Z^4+5)^2*(x^12-3*x^8+3*x^4
-1)^(1/2)*x^6+1024*RootOf(33554432*_Z^8-24576*_Z^4+5)^3*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+4096*RootOf(33554432*_Z
^8-24576*_Z^4+5)^4*x^4-3*x^8-32*RootOf(33554432*_Z^8-24576*_Z^4+5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-128*RootOf(3
3554432*_Z^8-24576*_Z^4+5)^2*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+3*x^4+16*RootOf(33554432*_Z^8-24576*_Z^4+5)*(x^12-
3*x^8+3*x^4-1)^(1/4)*x-1)/(4096*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x^4-x^4+1)/(8192*RootOf(33554432*_Z^8-245
76*_Z^4+5)^4*x-3*x-1)^2/(-1+x)^2/(8192*RootOf(33554432*_Z^8-24576*_Z^4+5)^4*x-3*x+1)^2/(1+x)^2))/(x^4-1)^(3/4)
*((x^4-1)^3)^(1/4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + 2)*(x^4 - 1)^(1/4)/((x^8 + 2*x^4 + 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^4-1\right )}^{1/4}\,\left (x^4+2\right )}{x^2\,\left (x^8+2\,x^4+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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