\(\int \frac {1+3 x^4+x^8}{x^2 (1+x^4)^{3/4} (1+3 x^4+3 x^8)} \, dx\) [1311]

Optimal result

Integrand size = 37, antiderivative size = 94 \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=-\frac {\sqrt [4]{1+x^4}}{x}-\frac {\arctan \left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{-x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}}+\frac {\text {arctanh}\left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}} \]

[Out]

-(x^4+1)^(1/4)/x-1/3*arctan(3^(1/2)*x*(x^4+1)^(1/4)/(-x^2+(x^4+1)^(1/2)))*3^(1/2)+1/3*arctanh(3^(1/2)*x*(x^4+1
)^(1/4)/(x^2+(x^4+1)^(1/2)))*3^(1/2)

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.69, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6860, 270, 1542, 508, 304, 211, 214} \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=-\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \arctan \left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )-\frac {\left (3-i \sqrt {3}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}}+\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} \sqrt [4]{x^4+1}}\right )+\frac {\left (3-i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}} x}{\sqrt [4]{x^4+1}}\right )}{6 \sqrt [4]{-\frac {-\sqrt {3}+3 i}{\sqrt {3}+3 i}}}-\frac {\sqrt [4]{x^4+1}}{x} \]

[In]

Int[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)),x]

[Out]

-((1 + x^4)^(1/4)/x) - ((3 + I*Sqrt[3])*(-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)*ArcTan[x/((-((3*I - Sqrt[3]
)/(3*I + Sqrt[3])))^(1/4)*(1 + x^4)^(1/4))])/6 - ((3 - I*Sqrt[3])*ArcTan[((-((3*I - Sqrt[3])/(3*I + Sqrt[3])))
^(1/4)*x)/(1 + x^4)^(1/4)])/(6*(-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)) + ((3 + I*Sqrt[3])*(-((3*I - Sqrt[3
])/(3*I + Sqrt[3])))^(1/4)*ArcTanh[x/((-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)*(1 + x^4)^(1/4))])/6 + ((3 -
I*Sqrt[3])*ArcTanh[((-((3*I - Sqrt[3])/(3*I + Sqrt[3])))^(1/4)*x)/(1 + x^4)^(1/4)])/(6*(-((3*I - Sqrt[3])/(3*I
+ Sqrt[3])))^(1/4))

Rule 211

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

Rule 214

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

Rule 270

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

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 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 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*} \text {integral}& = \int \left (\frac {1}{x^2 \left (1+x^4\right )^{3/4}}-\frac {2 x^6}{\left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )}\right ) \, dx \\ & = -\left (2 \int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx\right )+\int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1+x^4}}{x}-2 \int \left (\frac {i \left (-3+i \sqrt {3}\right ) x^2}{\sqrt {3} \left (-3+i \sqrt {3}-6 x^4\right ) \left (1+x^4\right )^{3/4}}-\frac {i \left (3+i \sqrt {3}\right ) x^2}{\sqrt {3} \left (1+x^4\right )^{3/4} \left (3+i \sqrt {3}+6 x^4\right )}\right ) \, dx \\ & = -\frac {\sqrt [4]{1+x^4}}{x}-\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (3+i \sqrt {3}+6 x^4\right )} \, dx+\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {x^2}{\left (-3+i \sqrt {3}-6 x^4\right ) \left (1+x^4\right )^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1+x^4}}{x}-\left (2 \left (1-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {x^2}{3+i \sqrt {3}-\left (-3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (2 \left (1+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {x^2}{-3+i \sqrt {3}-\left (3+i \sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = -\frac {\sqrt [4]{1+x^4}}{x}-\frac {\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3 i+\sqrt {3}}-\sqrt {-3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {-3 i+\sqrt {3}}}+\frac {\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3 i+\sqrt {3}}+\sqrt {-3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {-3 i+\sqrt {3}}}+\frac {\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-3 i+\sqrt {3}}-\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {3 i+\sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-3 i+\sqrt {3}}+\sqrt {3 i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {3 i+\sqrt {3}}} \\ & = -\frac {\sqrt [4]{1+x^4}}{x}-\frac {1}{2} \left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \arctan \left (\frac {x}{\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )-\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \arctan \left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}}}+\frac {1}{2} \left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \text {arctanh}\left (\frac {x}{\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{-\frac {3 i-\sqrt {3}}{3 i+\sqrt {3}}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=-\frac {\sqrt [4]{1+x^4}}{x}+\frac {\arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [4]{1+x^4}}\right )}{\sqrt {3}}+\frac {\text {arctanh}\left (\frac {\sqrt {3} x \sqrt [4]{1+x^4}}{x^2+\sqrt {1+x^4}}\right )}{\sqrt {3}} \]

[In]

Integrate[(1 + 3*x^4 + x^8)/(x^2*(1 + x^4)^(3/4)*(1 + 3*x^4 + 3*x^8)),x]

[Out]

-((1 + x^4)^(1/4)/x) + ArcTan[(Sqrt[3]*x)/(x - 2*(1 + x^4)^(1/4))]/Sqrt[3] - ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^
4)^(1/4))]/Sqrt[3] + ArcTanh[(Sqrt[3]*x*(1 + x^4)^(1/4))/(x^2 + Sqrt[1 + x^4])]/Sqrt[3]

Maple [A] (verified)

Time = 49.72 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45

method	result	size
pseudoelliptic	\(\frac {\sqrt {3}\, \ln \left (\frac {\sqrt {3}\, \left (x^{4}+1\right )^{\frac {1}{4}} x +x^{2}+\sqrt {x^{4}+1}}{x^{2}}\right ) x -\sqrt {3}\, \ln \left (\frac {-\sqrt {3}\, \left (x^{4}+1\right )^{\frac {1}{4}} x +x^{2}+\sqrt {x^{4}+1}}{x^{2}}\right ) x +2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}+1\right )^{\frac {1}{4}}+x \right ) \sqrt {3}}{3 x}\right ) x -2 \sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{4}+1\right )^{\frac {1}{4}}+x \right ) \sqrt {3}}{3 x}\right ) x -6 \left (x^{4}+1\right )^{\frac {1}{4}}}{6 x}\)	\(136\)
trager	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{6}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{8}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \sqrt {x^{4}+1}\, x^{2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}-6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{6}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{8}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x^{5}+18 x^{7} \left (x^{4}+1\right )^{\frac {1}{4}}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {x^{4}+1}\, x^{2}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{4}+6 \left (x^{4}+1\right )^{\frac {3}{4}} x +12 x^{3} \left (x^{4}+1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{3 x^{8}+3 x^{4}+1}\right )}{6}\)	\(287\)
risch	\(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{16}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{10}+27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{12}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}-42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{6}+28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{8}-12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}-30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{4}-6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{16}+18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{13}+12 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{10}-27 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{12}-18 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{7}+42 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{9}+18 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{8}-12 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {3}{4}} x^{3}+30 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x^{5}+6 \sqrt {x^{12}+3 x^{8}+3 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}-11 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}+6 \left (x^{12}+3 x^{8}+3 x^{4}+1\right )^{\frac {1}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (x^{4}+1\right )^{2} \left (3 x^{8}+3 x^{4}+1\right )}\right )}{6}\right ) {\left (\left (x^{4}+1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}+1\right )^{\frac {3}{4}}}\)	\(628\)

[In]

int((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/6*(3^(1/2)*ln((3^(1/2)*(x^4+1)^(1/4)*x+x^2+(x^4+1)^(1/2))/x^2)*x-3^(1/2)*ln((-3^(1/2)*(x^4+1)^(1/4)*x+x^2+(x
^4+1)^(1/2))/x^2)*x+2*3^(1/2)*arctan(1/3*(2*(x^4+1)^(1/4)+x)*3^(1/2)/x)*x-2*3^(1/2)*arctan(1/3*(-2*(x^4+1)^(1/
4)+x)*3^(1/2)/x)*x-6*(x^4+1)^(1/4))/x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (78) = 156\).

Time = 11.62 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.41 \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, {\left (\sqrt {3} {\left (3 \, x^{5} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} - \sqrt {3} {\left (3 \, x^{7} + 4 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}\right )}}{21 \, x^{8} + 21 \, x^{4} - 1}\right ) - \sqrt {3} x \log \left (-\frac {441 \, x^{16} + 882 \, x^{12} + 543 \, x^{8} + 102 \, x^{4} + 4 \, \sqrt {3} {\left (63 \, x^{13} + 78 \, x^{9} + 24 \, x^{5} + x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, \sqrt {3} {\left (63 \, x^{15} + 111 \, x^{11} + 57 \, x^{7} + 8 \, x^{3}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 24 \, {\left (18 \, x^{14} + 27 \, x^{10} + 11 \, x^{6} + x^{2}\right )} \sqrt {x^{4} + 1} + 1}{9 \, x^{16} + 18 \, x^{12} + 15 \, x^{8} + 6 \, x^{4} + 1}\right ) + 12 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{12 \, x} \]

[In]

integrate((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x, algorithm="fricas")

[Out]

-1/12*(2*sqrt(3)*x*arctan(2*(sqrt(3)*(3*x^5 - x)*(x^4 + 1)^(3/4) - sqrt(3)*(3*x^7 + 4*x^3)*(x^4 + 1)^(1/4))/(2
1*x^8 + 21*x^4 - 1)) - sqrt(3)*x*log(-(441*x^16 + 882*x^12 + 543*x^8 + 102*x^4 + 4*sqrt(3)*(63*x^13 + 78*x^9 +
24*x^5 + x)*(x^4 + 1)^(3/4) + 4*sqrt(3)*(63*x^15 + 111*x^11 + 57*x^7 + 8*x^3)*(x^4 + 1)^(1/4) + 24*(18*x^14 +
27*x^10 + 11*x^6 + x^2)*sqrt(x^4 + 1) + 1)/(9*x^16 + 18*x^12 + 15*x^8 + 6*x^4 + 1)) + 12*(x^4 + 1)^(1/4))/x

Sympy [F(-1)]

Timed out. \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**8+3*x**4+1)/x**2/(x**4+1)**(3/4)/(3*x**8+3*x**4+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\int { \frac {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x, algorithm="maxima")

[Out]

integrate((x^8 + 3*x^4 + 1)/((3*x^8 + 3*x^4 + 1)*(x^4 + 1)^(3/4)*x^2), x)

Giac [F]

\[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\int { \frac {x^{8} + 3 \, x^{4} + 1}{{\left (3 \, x^{8} + 3 \, x^{4} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]

[In]

integrate((x^8+3*x^4+1)/x^2/(x^4+1)^(3/4)/(3*x^8+3*x^4+1),x, algorithm="giac")

[Out]

integrate((x^8 + 3*x^4 + 1)/((3*x^8 + 3*x^4 + 1)*(x^4 + 1)^(3/4)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+3 x^4+x^8}{x^2 \left (1+x^4\right )^{3/4} \left (1+3 x^4+3 x^8\right )} \, dx=\int \frac {x^8+3\,x^4+1}{x^2\,{\left (x^4+1\right )}^{3/4}\,\left (3\,x^8+3\,x^4+1\right )} \,d x \]

[In]

int((3*x^4 + x^8 + 1)/(x^2*(x^4 + 1)^(3/4)*(3*x^4 + 3*x^8 + 1)),x)

[Out]

int((3*x^4 + x^8 + 1)/(x^2*(x^4 + 1)^(3/4)*(3*x^4 + 3*x^8 + 1)), x)