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

Optimal. Leaf size=113 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^5-\text {$\#$1}}\& \right ]-\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\& ,\frac {\log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^5-3 \text {$\#$1}}\& \right ] \]

________________________________________________________________________________________

Rubi [C]  time = 3.19, antiderivative size = 1033, normalized size of antiderivative = 9.14, number of steps used = 33, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2056, 6728, 1270, 1517, 240, 212, 206, 203, 1429, 377, 208, 205} \begin {gather*} -\frac {i \left (-2+\sqrt {-2-2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}}+\frac {i \left (-2+\sqrt {-2+2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}}-\frac {i \left (-2+\sqrt {-2-2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1-i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1-i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2-2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}}+\frac {i \left (-2+\sqrt {-2+2 i \sqrt {3}}\right )^{3/4} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{-2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{\sqrt {3} \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{x^4+x^2}}-\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt {x}}{\sqrt [8]{2 \left (-1+i \sqrt {3}\right )} \sqrt [4]{x^2+1}}\right )}{3 \left (2 \left (-1+i \sqrt {3}\right )\right )^{7/8} \sqrt [4]{2+\sqrt {-2+2 i \sqrt {3}}} \sqrt [4]{x^4+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*(-2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(3/4)*Sqrt[x]*(1 + x^2)^(1/4)*ArcTan[((-2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1
/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(Sqrt[3]*(2*(-1 - I*Sqrt[3]))^(7/8)*(x^2 + x^4)^(1
/4)) - ((3 - I*Sqrt[3])*Sqrt[x]*(1 + x^2)^(1/4)*ArcTan[((2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1
- I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(3*(2*(-1 - I*Sqrt[3]))^(7/8)*(2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*(x^2
+ x^4)^(1/4)) + (I*(-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(3/4)*Sqrt[x]*(1 + x^2)^(1/4)*ArcTan[((-2 + Sqrt[-2 + (2*I
)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(Sqrt[3]*(2*(-1 + I*Sqrt[3]))^(7/8)*
(x^2 + x^4)^(1/4)) - ((3 + I*Sqrt[3])*Sqrt[x]*(1 + x^2)^(1/4)*ArcTan[((2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)*Sqr
t[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(3*(2*(-1 + I*Sqrt[3]))^(7/8)*(2 + Sqrt[-2 + (2*I)*Sqrt[3
]])^(1/4)*(x^2 + x^4)^(1/4)) - (I*(-2 + Sqrt[-2 - (2*I)*Sqrt[3]])^(3/4)*Sqrt[x]*(1 + x^2)^(1/4)*ArcTanh[((-2 +
Sqrt[-2 - (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(Sqrt[3]*(2*(-1 - I*S
qrt[3]))^(7/8)*(x^2 + x^4)^(1/4)) - ((3 - I*Sqrt[3])*Sqrt[x]*(1 + x^2)^(1/4)*ArcTanh[((2 + Sqrt[-2 - (2*I)*Sqr
t[3]])^(1/4)*Sqrt[x])/((2*(-1 - I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(3*(2*(-1 - I*Sqrt[3]))^(7/8)*(2 + Sqrt[-
2 - (2*I)*Sqrt[3]])^(1/4)*(x^2 + x^4)^(1/4)) + (I*(-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(3/4)*Sqrt[x]*(1 + x^2)^(1/4
)*ArcTanh[((-2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(Sqrt
[3]*(2*(-1 + I*Sqrt[3]))^(7/8)*(x^2 + x^4)^(1/4)) - ((3 + I*Sqrt[3])*Sqrt[x]*(1 + x^2)^(1/4)*ArcTanh[((2 + Sqr
t[-2 + (2*I)*Sqrt[3]])^(1/4)*Sqrt[x])/((2*(-1 + I*Sqrt[3]))^(1/8)*(1 + x^2)^(1/4))])/(3*(2*(-1 + I*Sqrt[3]))^(
7/8)*(2 + Sqrt[-2 + (2*I)*Sqrt[3]])^(1/4)*(x^2 + 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 205

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

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/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 1270

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominat
or[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + (e*x^(2*k))/f)^q*(a + (c*x^(4*k))/f)^p, x], x, (f*x)^(1/k)]
, x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]

Rule 1429

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

Rule 1517

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

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

________________________________________________________________________________________

Mathematica [F]  time = 3.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\sqrt [4]{x^2+x^4} \left (1+x^4+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.31, size = 113, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 149.23, size = 5894, normalized size = 52.16

method	result	size
trager	\(\text {Expression too large to display}\)	\(5894\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 \, {\left (4 \, x^{5} + x^{3} - 3 \, x\right )} x^{\frac {7}{2}}}{21 \, {\left (x^{8} + x^{4} + 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} - \int \frac {8 \, {\left ({\left (4 \, x^{4} + x^{2} - 3\right )} x^{\frac {15}{2}} + 2 \, {\left (4 \, x^{4} + x^{2} - 3\right )} x^{\frac {7}{2}}\right )}}{21 \, {\left (x^{16} + 2 \, x^{12} + 3 \, x^{8} + 2 \, x^{4} + 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(4*x^5 + x^3 - 3*x)*x^(7/2)/((x^8 + x^4 + 1)*(x^2 + 1)^(1/4)) - integrate(8/21*((4*x^4 + x^2 - 3)*x^(15/2
) + 2*(4*x^4 + x^2 - 3)*x^(7/2))/((x^16 + 2*x^12 + 3*x^8 + 2*x^4 + 1)*(x^2 + 1)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^8+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4/((x**2*(x**2 + 1))**(1/4)*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 1)), x)

________________________________________________________________________________________