\(\int \frac {\sqrt {-2+3 x^4} (-1+3 x^4)}{x^3 (1-3 x^4+3 x^8)} \, dx\) [2834]

Optimal result

Integrand size = 36, antiderivative size = 288 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \sqrt {3} \text {RootSum}\left [4+4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )-\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\frac {1}{2} \sqrt {3} \text {RootSum}\left [4-4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {6860, 281, 283, 223, 212, 1504, 1307, 1706, 385, 211, 214} \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=-\frac {\left (-\sqrt {3}+i\right ) \sqrt {\frac {3 \left (-\sqrt {3}+3 i\right )}{\sqrt {3}+i}} \arctan \left (\frac {x^2}{\sqrt {\frac {-\sqrt {3}+3 i}{3 \left (\sqrt {3}+i\right )}} \sqrt {3 x^4-2}}\right )}{2 \left (-\sqrt {3}+3 i\right )}-\frac {\left (\sqrt {3}+i\right ) \text {arctanh}\left (\frac {\sqrt {-\frac {3 \left (-\sqrt {3}+i\right )}{\sqrt {3}+3 i}} x^2}{\sqrt {3 x^4-2}}\right )}{2 \sqrt {\frac {2}{3} \left (3+i \sqrt {3}\right )}}+\frac {\sqrt {3 x^4-2}}{2 x^2} \]

[In]

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

[Out]

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

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 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 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 385

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 1307

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[e
*(f^2/c), Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1)*(S
imp[a*e - (c*d - b*e)*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]

Rule 1504

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^(2*(n/k
)))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[n, 0] && IntegerQ[m]

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

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 {\sqrt {-2+3 x^4}}{x^3}+\frac {3 x^5 \sqrt {-2+3 x^4}}{1-3 x^4+3 x^8}\right ) \, dx \\ & = 3 \int \frac {x^5 \sqrt {-2+3 x^4}}{1-3 x^4+3 x^8} \, dx-\int \frac {\sqrt {-2+3 x^4}}{x^3} \, dx \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-2+3 x^2}}{x^2} \, dx,x,x^2\right )\right )+\frac {3}{2} \text {Subst}\left (\int \frac {x^2 \sqrt {-2+3 x^2}}{1-3 x^2+3 x^4} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {1}{2} \text {Subst}\left (\int \frac {3-3 x^2}{\sqrt {-2+3 x^2} \left (1-3 x^2+3 x^4\right )} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {1}{2} \text {Subst}\left (\int \left (\frac {-3-3 i \sqrt {3}}{\sqrt {-2+3 x^2} \left (-3-i \sqrt {3}+6 x^2\right )}+\frac {-3+3 i \sqrt {3}}{\sqrt {-2+3 x^2} \left (-3+i \sqrt {3}+6 x^2\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \left (3 \left (1-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-2+3 x^2} \left (-3+i \sqrt {3}+6 x^2\right )} \, dx,x,x^2\right )+\frac {1}{2} \left (3 \left (1+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-2+3 x^2} \left (-3-i \sqrt {3}+6 x^2\right )} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-2+3 x^4}}{2 x^2}+\frac {1}{2} \left (3 \left (1-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{-3+i \sqrt {3}-\left (12+3 \left (-3+i \sqrt {3}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-2+3 x^4}}\right )+\frac {1}{2} \left (3 \left (1+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-i \sqrt {3}-\left (12+3 \left (-3-i \sqrt {3}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-2+3 x^4}}\right ) \\ & = \frac {\sqrt {-2+3 x^4}}{2 x^2}-\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {3 \left (3 i-\sqrt {3}\right )}{i+\sqrt {3}}} \arctan \left (\frac {x^2}{\sqrt {\frac {3 i-\sqrt {3}}{3 \left (i+\sqrt {3}\right )}} \sqrt {-2+3 x^4}}\right )}{2 \left (3 i-\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt {-\frac {3 \left (i-\sqrt {3}\right )}{3 i+\sqrt {3}}} x^2}{\sqrt {-2+3 x^4}}\right )}{2 \sqrt {\frac {2}{3} \left (3+i \sqrt {3}\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\frac {1}{2} \left (\frac {\sqrt {-2+3 x^4}}{x^2}+\sqrt {3} \text {RootSum}\left [4+4 \text {$\#$1}-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )-\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\sqrt {3} \text {RootSum}\left [4-4 \text {$\#$1}+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right )+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {3} x^2+\sqrt {-2+3 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]\right ) \]

[In]

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

[Out]

(Sqrt[-2 + 3*x^4]/x^2 + Sqrt[3]*RootSum[4 + 4*#1 - 2*#1^3 + #1^4 & , (-2*Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] -
#1] - Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(2 - 3*#1^2
+ 2*#1^3) & ] - Sqrt[3]*RootSum[4 - 4*#1 + 2*#1^3 + #1^4 & , (-2*Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1] + L
og[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1 + Log[Sqrt[3]*x^2 + Sqrt[-2 + 3*x^4] - #1]*#1^2)/(-2 + 3*#1^2 + 2*#
1^3) & ])/2

Maple [N/A] (verified)

Time = 7.01 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {2}\, \left (-\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {3}+\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}-3 x^{2}}{3 x^{2}}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}+3 x^{2}}{3 x^{2}}\right )\right ) 3^{\frac {1}{4}}}{16}\)	\(244\)
elliptic	\(\frac {\sqrt {3 x^{4}-2}}{2 x^{2}}-\frac {\sqrt {2}\, \left (-\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {3}+\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}-3 x^{2}}{3 x^{2}}\right )+\left (-2 \sqrt {3}+2\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}+3 x^{2}}{3 x^{2}}\right )\right ) 3^{\frac {1}{4}}}{16}\)	\(244\)
default	\(-\frac {\left (3 x^{4}-2\right )^{\frac {3}{2}}}{4 x^{2}}+\frac {3 x^{2} \sqrt {3 x^{4}-2}}{4}-\frac {\ln \left (\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}\right ) \sqrt {3}}{2}+\frac {\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {2}\, 3^{\frac {3}{4}}}{16}-\frac {\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) 3^{\frac {1}{4}} \sqrt {2}}{16}-\frac {\sqrt {2}\, \left (3^{\frac {1}{4}}-3^{\frac {3}{4}}\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}-3 x^{2}}{3 x^{2}}\right )}{8}-\frac {\sqrt {2}\, \left (3^{\frac {1}{4}}-3^{\frac {3}{4}}\right ) \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {3 x^{4}-2}+3 x^{2}}{3 x^{2}}\right )}{8}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {3}\, \sqrt {3 x^{4}-2}}{3 x^{2}}\right )}{2}\)	\(318\)
pseudoelliptic	\(-\frac {\sqrt {3}\, \left (-\frac {\ln \left (\frac {-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) 3^{\frac {1}{4}} \sqrt {2}\, x^{2}}{2}+\frac {\ln \left (\frac {3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}{-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}\, x^{2}+\sqrt {3}\, x^{4}+3 x^{4}-2}\right ) \sqrt {2}\, 3^{\frac {3}{4}} x^{2}}{6}-x^{2} \left (3^{\frac {1}{4}}-\frac {3^{\frac {3}{4}}}{3}\right ) \sqrt {2}\, \arctan \left (\frac {\left (3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}+\sqrt {3}\, x^{2}\right ) \sqrt {3}}{3 x^{2}}\right )+2 \ln \left (\frac {\sqrt {3 x^{4}-2}-\sqrt {3}\, x^{2}}{x^{2}}\right ) x^{2}-2 \ln \left (\frac {\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}}{x^{2}}\right ) x^{2}+4 \ln \left (\sqrt {3}\, x^{2}+\sqrt {3 x^{4}-2}\right ) x^{2}-\frac {4 \sqrt {3}\, \sqrt {3 x^{4}-2}}{3}+x^{2} \left (3^{\frac {1}{4}}-\frac {3^{\frac {3}{4}}}{3}\right ) \arctan \left (\frac {\left (-3^{\frac {1}{4}} \sqrt {3 x^{4}-2}\, \sqrt {2}+\sqrt {3}\, x^{2}\right ) \sqrt {3}}{3 x^{2}}\right ) \sqrt {2}\right )}{8 x^{2}}\)	\(366\)
trager	\(\text {Expression too large to display}\)	\(730\)

[In]

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

[Out]

1/2/x^2*(3*x^4-2)^(1/2)-1/16*2^(1/2)*(-ln((-3^(1/4)*(3*x^4-2)^(1/2)*2^(1/2)*x^2+3^(1/2)*x^4+3*x^4-2)/(3^(1/4)*
(3*x^4-2)^(1/2)*2^(1/2)*x^2+3^(1/2)*x^4+3*x^4-2))*3^(1/2)+ln((3^(1/4)*(3*x^4-2)^(1/2)*2^(1/2)*x^2+3^(1/2)*x^4+
3*x^4-2)/(-3^(1/4)*(3*x^4-2)^(1/2)*2^(1/2)*x^2+3^(1/2)*x^4+3*x^4-2))+(-2*3^(1/2)+2)*arctan(1/3*(2^(1/2)*3^(3/4
)*(3*x^4-2)^(1/2)-3*x^2)/x^2)+(-2*3^(1/2)+2)*arctan(1/3*(2^(1/2)*3^(3/4)*(3*x^4-2)^(1/2)+3*x^2)/x^2))*3^(1/4)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.27 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=-\frac {\sqrt {2} x^{2} \sqrt {-\sqrt {-3} + 3} \log \left (\frac {2 \, \sqrt {-3} x^{4} + 6 \, x^{4} + \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} + x^{2}\right )} \sqrt {-\sqrt {-3} + 3} - 4}{x^{4}}\right ) - \sqrt {2} x^{2} \sqrt {-\sqrt {-3} + 3} \log \left (\frac {2 \, \sqrt {-3} x^{4} + 6 \, x^{4} - \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} + x^{2}\right )} \sqrt {-\sqrt {-3} + 3} - 4}{x^{4}}\right ) + \sqrt {2} x^{2} \sqrt {\sqrt {-3} + 3} \log \left (-\frac {2 \, \sqrt {-3} x^{4} - 6 \, x^{4} + \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} - x^{2}\right )} \sqrt {\sqrt {-3} + 3} + 4}{x^{4}}\right ) - \sqrt {2} x^{2} \sqrt {\sqrt {-3} + 3} \log \left (-\frac {2 \, \sqrt {-3} x^{4} - 6 \, x^{4} - \sqrt {2} \sqrt {3 \, x^{4} - 2} {\left (\sqrt {-3} x^{2} - x^{2}\right )} \sqrt {\sqrt {-3} + 3} + 4}{x^{4}}\right ) - 8 \, \sqrt {3 \, x^{4} - 2}}{16 \, x^{2}} \]

[In]

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

[Out]

-1/16*(sqrt(2)*x^2*sqrt(-sqrt(-3) + 3)*log((2*sqrt(-3)*x^4 + 6*x^4 + sqrt(2)*sqrt(3*x^4 - 2)*(sqrt(-3)*x^2 + x
^2)*sqrt(-sqrt(-3) + 3) - 4)/x^4) - sqrt(2)*x^2*sqrt(-sqrt(-3) + 3)*log((2*sqrt(-3)*x^4 + 6*x^4 - sqrt(2)*sqrt
(3*x^4 - 2)*(sqrt(-3)*x^2 + x^2)*sqrt(-sqrt(-3) + 3) - 4)/x^4) + sqrt(2)*x^2*sqrt(sqrt(-3) + 3)*log(-(2*sqrt(-
3)*x^4 - 6*x^4 + sqrt(2)*sqrt(3*x^4 - 2)*(sqrt(-3)*x^2 - x^2)*sqrt(sqrt(-3) + 3) + 4)/x^4) - sqrt(2)*x^2*sqrt(
sqrt(-3) + 3)*log(-(2*sqrt(-3)*x^4 - 6*x^4 - sqrt(2)*sqrt(3*x^4 - 2)*(sqrt(-3)*x^2 - x^2)*sqrt(sqrt(-3) + 3) +
4)/x^4) - 8*sqrt(3*x^4 - 2))/x^2

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\int { \frac {{\left (3 \, x^{4} - 1\right )} \sqrt {3 \, x^{4} - 2}}{{\left (3 \, x^{8} - 3 \, x^{4} + 1\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\frac {1}{16} \, \sqrt {3} {\left ({\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} - 8 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 4\right )} \log \left ({\left | -{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{8} + 4 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{6} - 24 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{4} + 16 \, {\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} - 16 \right |}\right ) + \frac {2 \, \sqrt {3}}{{\left (\sqrt {3} x^{2} - \sqrt {3 \, x^{4} - 2}\right )}^{2} + 2} \]

[In]

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

[Out]

1/16*sqrt(3)*((sqrt(3)*x^2 - sqrt(3*x^4 - 2))^4 - 8*(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^2 + 4)*log(abs(-(sqrt(3)*x
^2 - sqrt(3*x^4 - 2))^8 + 4*(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^6 - 24*(sqrt(3)*x^2 - sqrt(3*x^4 - 2))^4 + 16*(sqr
t(3)*x^2 - sqrt(3*x^4 - 2))^2 - 16)) + 2*sqrt(3)/((sqrt(3)*x^2 - sqrt(3*x^4 - 2))^2 + 2)

Mupad [N/A]

Not integrable

Time = 7.41 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.12 \[ \int \frac {\sqrt {-2+3 x^4} \left (-1+3 x^4\right )}{x^3 \left (1-3 x^4+3 x^8\right )} \, dx=\int \frac {\left (3\,x^4-1\right )\,\sqrt {3\,x^4-2}}{x^3\,\left (3\,x^8-3\,x^4+1\right )} \,d x \]

[In]

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

[Out]

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