3.15.28 \(\int \frac {\sqrt [4]{1+x^4} (2+x^4)}{x^2 (-1+2 x^4+x^8)} \, dx\) [1428]

Optimal. Leaf size=101 \[ \frac {2 \sqrt [4]{1+x^4}}{x}-\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-4 \log (x)+4 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right )+7 \log (x) \text {$\#$1}^4-7 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(425\) vs. \(2(101)=202\).
time = 0.94, antiderivative size = 425, normalized size of antiderivative = 4.21, number of steps used = 40, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6860, 283, 338, 304, 209, 212, 1542, 524, 1532, 508} \begin {gather*} -\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+\frac {1}{4} \sqrt [4]{10+7 \sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \sqrt [4]{10-7 \sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {2 \sqrt [4]{x^4+1}}{x}-\frac {1}{4} \sqrt [4]{10+7 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{10-7 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + x^4)^(1/4))/x - (5*(2 - Sqrt[2])*x^3*AppellF1[3/4, -1/4, 1, 7/4, -x^4, (1 - Sqrt[2])*x^4])/12 - (5*(2
+ Sqrt[2])*x^3*AppellF1[3/4, 1, -1/4, 7/4, (1 + Sqrt[2])*x^4, -x^4])/12 + ((2 - Sqrt[2])^(1/4)*ArcTan[((2 - Sq
rt[2])^(1/4)*x)/(1 + x^4)^(1/4)])/4 + ((10 + 7*Sqrt[2])^(1/4)*ArcTan[((2 - Sqrt[2])^(1/4)*x)/(1 + x^4)^(1/4)])
/4 - ((10 - 7*Sqrt[2])^(1/4)*ArcTan[((2 + Sqrt[2])^(1/4)*x)/(1 + x^4)^(1/4)])/4 + ((2 + Sqrt[2])^(1/4)*ArcTan[
((2 + Sqrt[2])^(1/4)*x)/(1 + x^4)^(1/4)])/4 - ((2 - Sqrt[2])^(1/4)*ArcTanh[((2 - Sqrt[2])^(1/4)*x)/(1 + x^4)^(
1/4)])/4 - ((10 + 7*Sqrt[2])^(1/4)*ArcTanh[((2 - Sqrt[2])^(1/4)*x)/(1 + x^4)^(1/4)])/4 + ((10 - 7*Sqrt[2])^(1/
4)*ArcTanh[((2 + Sqrt[2])^(1/4)*x)/(1 + x^4)^(1/4)])/4 - ((2 + Sqrt[2])^(1/4)*ArcTanh[((2 + Sqrt[2])^(1/4)*x)/
(1 + x^4)^(1/4)])/4

Rule 209

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

Rule 212

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

Rule 283

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 304

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 338

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 508

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 1532

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 1542

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 6860

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 (-1+2 x^4+x^8\right )} \, dx &=\int \left (-\frac {2 \sqrt [4]{1+x^4}}{x^2}+\frac {x^2 \sqrt [4]{1+x^4} \left (5+2 x^4\right )}{-1+2 x^4+x^8}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1+x^4}}{x^2} \, dx\right )+\int \frac {x^2 \sqrt [4]{1+x^4} \left (5+2 x^4\right )}{-1+2 x^4+x^8} \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-2 \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx+\int \left (\frac {5 x^2 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8}+\frac {2 x^6 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8}\right ) \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}+2 \int \frac {x^6 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8} \, dx-2 \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+5 \int \frac {x^2 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8} \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}+2 \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx-2 \int \frac {x^2 \left (-1+x^4\right )}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )} \, dx+5 \int \left (-\frac {x^2 \sqrt [4]{1+x^4}}{\sqrt {2} \left (-2+2 \sqrt {2}-2 x^4\right )}-\frac {x^2 \sqrt [4]{1+x^4}}{\sqrt {2} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-2 \int \left (-\frac {x^2}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )}+\frac {x^6}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )}\right ) \, dx+2 \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {5 \int \frac {x^2 \sqrt [4]{1+x^4}}{-2+2 \sqrt {2}-2 x^4} \, dx}{\sqrt {2}}-\frac {5 \int \frac {x^2 \sqrt [4]{1+x^4}}{2+2 \sqrt {2}+2 x^4} \, dx}{\sqrt {2}}\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+2 \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )} \, dx-2 \int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )} \, dx+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+2 \int \left (-\frac {x^2}{\sqrt {2} \left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}}-\frac {x^2}{\sqrt {2} \left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx-2 \int \left (-\frac {\left (-2+2 \sqrt {2}\right ) x^2}{2 \sqrt {2} \left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {\left (2+2 \sqrt {2}\right ) x^2}{2 \sqrt {2} \left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )-\sqrt {2} \int \frac {x^2}{\left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}} \, dx-\sqrt {2} \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )} \, dx-\left (-2+\sqrt {2}\right ) \int \frac {x^2}{\left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}} \, dx-\left (2+\sqrt {2}\right ) \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )} \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )-\sqrt {2} \text {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \text {Subst}\left (\int \frac {x^2}{2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {x^2}{2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}-\frac {1}{4} \left (2-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{4} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )+\frac {\left (2+\sqrt {2}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\left (2-\sqrt {2}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {\left (2+\sqrt {2}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {1}{4} \sqrt [4]{10-7 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 101, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{1+x^4}}{x}-\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-4 \log (x)+4 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right )+7 \log (x) \text {$\#$1}^4-7 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 33.87, size = 5218, normalized size = 51.66 \[\text {output too large to display}\]

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^4-1),x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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^4-1),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 5.73, size = 1214, normalized size = 12.02 \begin {gather*} \frac {4 \, \sqrt {2} x {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (\sqrt {2} {\left (69 \, x^{8} + 18 \, x^{4} + 2 \, {\left (1451 \, x^{6} - 601 \, x^{2} - \sqrt {2} {\left (1026 \, x^{6} - 425 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} \sqrt {1393 \, \sqrt {2} + 1970} - \sqrt {2} {\left (47 \, x^{8} + 8 \, x^{4} - 13\right )} - 17\right )} \sqrt {-{\left (2671 \, \sqrt {2} - 3778\right )} \sqrt {1393 \, \sqrt {2} + 1970}} - 196 \, {\left (7 \, x^{5} - \sqrt {2} {\left (5 \, x^{5} - 2 \, x\right )} - 3 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} - 196 \, {\left (239 \, x^{7} - 99 \, x^{3} - \sqrt {2} {\left (169 \, x^{7} - 70 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {1393 \, \sqrt {2} + 1970}\right )} {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} + 2 \, x^{4} - 1\right )}}\right ) + 4 \, \sqrt {2} x {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (69 \, x^{8} + 18 \, x^{4} + 2 \, {\left (1451 \, x^{6} - 601 \, x^{2} + \sqrt {2} {\left (1026 \, x^{6} - 425 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} \sqrt {-1393 \, \sqrt {2} + 1970} + \sqrt {2} {\left (47 \, x^{8} + 8 \, x^{4} - 13\right )} - 17\right )} \sqrt {{\left (2671 \, \sqrt {2} + 3778\right )} \sqrt {-1393 \, \sqrt {2} + 1970}} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} - 196 \, {\left ({\left (7 \, x^{5} + \sqrt {2} {\left (5 \, x^{5} - 2 \, x\right )} - 3 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + {\left (239 \, x^{7} - 99 \, x^{3} + \sqrt {2} {\left (169 \, x^{7} - 70 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {-1393 \, \sqrt {2} + 1970}\right )} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} + 2 \, x^{4} - 1\right )}}\right ) + \sqrt {2} x {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} + \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {-1393 \, \sqrt {2} + 1970} + {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} + \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} + \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {-1393 \, \sqrt {2} + 1970}\right )} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) - \sqrt {2} x {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} + \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {-1393 \, \sqrt {2} + 1970} - {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} + \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} + \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {-1393 \, \sqrt {2} + 1970}\right )} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) - \sqrt {2} x {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} - \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {1393 \, \sqrt {2} + 1970} + {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} - \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} - \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {1393 \, \sqrt {2} + 1970}\right )} {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) + \sqrt {2} x {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} - \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {1393 \, \sqrt {2} + 1970} - {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} - \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} - \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {1393 \, \sqrt {2} + 1970}\right )} {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) + 64 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{32 \, 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^4-1),x, algorithm="fricas")

[Out]

1/32*(4*sqrt(2)*x*(1393*sqrt(2) + 1970)^(1/4)*arctan(-1/98*(sqrt(2)*(69*x^8 + 18*x^4 + 2*(1451*x^6 - 601*x^2 -
sqrt(2)*(1026*x^6 - 425*x^2))*sqrt(x^4 + 1)*sqrt(1393*sqrt(2) + 1970) - sqrt(2)*(47*x^8 + 8*x^4 - 13) - 17)*s
qrt(-(2671*sqrt(2) - 3778)*sqrt(1393*sqrt(2) + 1970)) - 196*(7*x^5 - sqrt(2)*(5*x^5 - 2*x) - 3*x)*(x^4 + 1)^(3
/4) - 196*(239*x^7 - 99*x^3 - sqrt(2)*(169*x^7 - 70*x^3))*(x^4 + 1)^(1/4)*sqrt(1393*sqrt(2) + 1970))*(1393*sqr
t(2) + 1970)^(1/4)/(x^8 + 2*x^4 - 1)) + 4*sqrt(2)*x*(-1393*sqrt(2) + 1970)^(1/4)*arctan(1/98*(sqrt(2)*(69*x^8
+ 18*x^4 + 2*(1451*x^6 - 601*x^2 + sqrt(2)*(1026*x^6 - 425*x^2))*sqrt(x^4 + 1)*sqrt(-1393*sqrt(2) + 1970) + sq
rt(2)*(47*x^8 + 8*x^4 - 13) - 17)*sqrt((2671*sqrt(2) + 3778)*sqrt(-1393*sqrt(2) + 1970))*(-1393*sqrt(2) + 1970
)^(1/4) - 196*((7*x^5 + sqrt(2)*(5*x^5 - 2*x) - 3*x)*(x^4 + 1)^(3/4) + (239*x^7 - 99*x^3 + sqrt(2)*(169*x^7 -
70*x^3))*(x^4 + 1)^(1/4)*sqrt(-1393*sqrt(2) + 1970))*(-1393*sqrt(2) + 1970)^(1/4))/(x^8 + 2*x^4 - 1)) + sqrt(2
)*x*(-1393*sqrt(2) + 1970)^(1/4)*log((4*(11*x^5 + sqrt(2)*(6*x^5 - 5*x) - x)*(x^4 + 1)^(3/4) + 4*(331*x^7 - 13
7*x^3 + sqrt(2)*(234*x^7 - 97*x^3))*(x^4 + 1)^(1/4)*sqrt(-1393*sqrt(2) + 1970) + (4*(57*x^6 - 23*x^2 + sqrt(2)
*(40*x^6 - 17*x^2))*sqrt(x^4 + 1) + (10444*x^8 + 2260*x^4 + sqrt(2)*(7385*x^8 + 1598*x^4 - 1929) - 2728)*sqrt(
-1393*sqrt(2) + 1970))*(-1393*sqrt(2) + 1970)^(1/4))/(x^8 + 2*x^4 - 1)) - sqrt(2)*x*(-1393*sqrt(2) + 1970)^(1/
4)*log((4*(11*x^5 + sqrt(2)*(6*x^5 - 5*x) - x)*(x^4 + 1)^(3/4) + 4*(331*x^7 - 137*x^3 + sqrt(2)*(234*x^7 - 97*
x^3))*(x^4 + 1)^(1/4)*sqrt(-1393*sqrt(2) + 1970) - (4*(57*x^6 - 23*x^2 + sqrt(2)*(40*x^6 - 17*x^2))*sqrt(x^4 +
1) + (10444*x^8 + 2260*x^4 + sqrt(2)*(7385*x^8 + 1598*x^4 - 1929) - 2728)*sqrt(-1393*sqrt(2) + 1970))*(-1393*
sqrt(2) + 1970)^(1/4))/(x^8 + 2*x^4 - 1)) - sqrt(2)*x*(1393*sqrt(2) + 1970)^(1/4)*log((4*(11*x^5 - sqrt(2)*(6*
x^5 - 5*x) - x)*(x^4 + 1)^(3/4) + 4*(331*x^7 - 137*x^3 - sqrt(2)*(234*x^7 - 97*x^3))*(x^4 + 1)^(1/4)*sqrt(1393
*sqrt(2) + 1970) + (4*(57*x^6 - 23*x^2 - sqrt(2)*(40*x^6 - 17*x^2))*sqrt(x^4 + 1) + (10444*x^8 + 2260*x^4 - sq
rt(2)*(7385*x^8 + 1598*x^4 - 1929) - 2728)*sqrt(1393*sqrt(2) + 1970))*(1393*sqrt(2) + 1970)^(1/4))/(x^8 + 2*x^
4 - 1)) + sqrt(2)*x*(1393*sqrt(2) + 1970)^(1/4)*log((4*(11*x^5 - sqrt(2)*(6*x^5 - 5*x) - x)*(x^4 + 1)^(3/4) +
4*(331*x^7 - 137*x^3 - sqrt(2)*(234*x^7 - 97*x^3))*(x^4 + 1)^(1/4)*sqrt(1393*sqrt(2) + 1970) - (4*(57*x^6 - 23
*x^2 - sqrt(2)*(40*x^6 - 17*x^2))*sqrt(x^4 + 1) + (10444*x^8 + 2260*x^4 - sqrt(2)*(7385*x^8 + 1598*x^4 - 1929)
- 2728)*sqrt(1393*sqrt(2) + 1970))*(1393*sqrt(2) + 1970)^(1/4))/(x^8 + 2*x^4 - 1)) + 64*(x^4 + 1)^(1/4))/x

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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**4-1),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^4-1),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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^4-1\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 - 1)),x)

[Out]

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

________________________________________________________________________________________