\(\int \frac {\sqrt {1-3 x^2-2 x^4} (1+2 x^4)}{(-1+x^2+2 x^4) (-1+2 x^2+2 x^4)} \, dx\) [2159]

Optimal result

Integrand size = 50, antiderivative size = 158 \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=\arctan \left (\frac {x \sqrt {1-3 x^2-2 x^4}}{-1+3 x^2+2 x^4}\right )-i \sqrt {2} \text {arctanh}\left (\frac {-2 i x+2 \sqrt {2} x^3-2 i x \sqrt {1-3 x^2-2 x^4}}{-\sqrt {2}+3 \sqrt {2} x^2+2 \sqrt {2} x^4-\sqrt {2} \sqrt {1-3 x^2-2 x^4}-2 i x^2 \sqrt {1-3 x^2-2 x^4}}\right ) \]

[Out]

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

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.97 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.96, number of steps used = 32, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6860, 1222, 1194, 538, 435, 430, 1226, 551} \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=-\sqrt {\frac {1}{2} \left (9 \sqrt {17}-37\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {3 \sqrt {17}-5} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {\left (1+2 \sqrt {3}-\sqrt {17}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {\left (1-2 \sqrt {3}-\sqrt {17}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-3+\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right ) \]

[In]

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

[Out]

((1 - 2*Sqrt[3] - Sqrt[17])*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4])/Sqrt[2*(3 + Sq
rt[17])] + ((1 + 2*Sqrt[3] - Sqrt[17])*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4])/Sqr
t[2*(3 + Sqrt[17])] + (Sqrt[-5 + 3*Sqrt[17]]*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4
])/2 - Sqrt[(-37 + 9*Sqrt[17])/2]*EllipticF[ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4] + 2*Sqrt[
2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/4, ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sqrt[17])/4] - Sqrt
[2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/(2*(1 - Sqrt[3])), ArcSin[(2*x)/Sqrt[-3 + Sqrt[17]]], (-13 + 3*Sq
rt[17])/4] - Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(3 - Sqrt[17])/(2*(1 + Sqrt[3])), ArcSin[(2*x)/Sqrt[-3 + Sqrt[1
7]]], (-13 + 3*Sqrt[17])/4] + 2*Sqrt[2/(3 + Sqrt[17])]*EllipticPi[(-3 + Sqrt[17])/2, ArcSin[(2*x)/Sqrt[-3 + Sq
rt[17]]], (-13 + 3*Sqrt[17])/4]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 1194

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[2*Sqrt[-c], Int[(d + e*x^2)/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c,
d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]

Rule 1222

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-(e^2)^(-1), Int[(c*d -
b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 - b*d*e + a*e^2)/e^2, Int[(a + b*x^2 + c*x^4
)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && IGtQ[p + 1/2, 0]

Rule 1226

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[2*Sqrt[-c], Int[1/((d + e*x^2)*Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a,
b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]

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 {1-3 x^2-2 x^4}}{1+x^2}+\frac {2 \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2}-\frac {2 \left (1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2+2 x^4}\right ) \, dx \\ & = 2 \int \frac {\sqrt {1-3 x^2-2 x^4}}{-1+2 x^2} \, dx-2 \int \frac {\left (1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2+2 x^4} \, dx+\int \frac {\sqrt {1-3 x^2-2 x^4}}{1+x^2} \, dx \\ & = -\left (\frac {1}{2} \int \frac {8+4 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx\right )+2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-2 \int \frac {1}{\left (-1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-2 \int \left (\frac {2 \sqrt {1-3 x^2-2 x^4}}{2-2 \sqrt {3}+4 x^2}+\frac {2 \sqrt {1-3 x^2-2 x^4}}{2+2 \sqrt {3}+4 x^2}\right ) \, dx-\int \frac {1+2 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx \\ & = -\left (4 \int \frac {\sqrt {1-3 x^2-2 x^4}}{2-2 \sqrt {3}+4 x^2} \, dx\right )-4 \int \frac {\sqrt {1-3 x^2-2 x^4}}{2+2 \sqrt {3}+4 x^2} \, dx-\sqrt {2} \int \frac {8+4 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (2 \sqrt {2}\right ) \int \frac {1+2 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (1+x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (-1+2 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx \\ & = 2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-3+\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{4} \int \frac {12-2 \left (2-2 \sqrt {3}\right )+8 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx+\frac {1}{4} \int \frac {12-2 \left (2+2 \sqrt {3}\right )+8 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx-2 \left (\sqrt {2} \int \frac {\sqrt {3+\sqrt {17}+4 x^2}}{\sqrt {-3+\sqrt {17}-4 x^2}} \, dx\right )-\left (2 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\left (2-2 \sqrt {3}+4 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-\left (2 \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\left (2+2 \sqrt {3}+4 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-\left (\sqrt {2} \left (5-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (\sqrt {2} \left (1+\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx \\ & = -\sqrt {2 \left (3+\sqrt {17}\right )} E\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-3+\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {\int \frac {12-2 \left (2-2 \sqrt {3}\right )+8 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx}{\sqrt {2}}+\frac {\int \frac {12-2 \left (2+2 \sqrt {3}\right )+8 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx}{\sqrt {2}}-\left (4 \sqrt {2} \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (2-2 \sqrt {3}+4 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (4 \sqrt {2} \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (2+2 \sqrt {3}+4 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx \\ & = -\sqrt {2 \left (3+\sqrt {17}\right )} E\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-3+\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \left (\sqrt {2} \int \frac {\sqrt {3+\sqrt {17}+4 x^2}}{\sqrt {-3+\sqrt {17}-4 x^2}} \, dx\right )+\left (\sqrt {2} \left (1-2 \sqrt {3}-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (\sqrt {2} \left (1+2 \sqrt {3}-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx \\ & = \frac {\left (1-2 \sqrt {3}-\sqrt {17}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {\left (1+2 \sqrt {3}-\sqrt {17}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (3-\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )},\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-3+\sqrt {17}\right ),\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right ),\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {1-3 x^2-2 x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {1-3 x^2-2 x^4}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 5.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.35

method	result	size
elliptic	\(\frac {\left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}}{x}\right )-2 \arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}\, \sqrt {2}}{2 x}\right )\right ) \sqrt {2}}{2}\)	\(55\)
default	\(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 i x \sqrt {2}-2 i x^{2}-\sqrt {2}\, x^{2}-i+\sqrt {2}-3 x \right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {4 i x +2 x^{2}-2+\left (2 i x^{2}-3 x +i\right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}+\arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}}{x}\right )\)	\(125\)
pseudoelliptic	\(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 i x \sqrt {2}-2 i x^{2}-\sqrt {2}\, x^{2}-i+\sqrt {2}-3 x \right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {4 i x +2 x^{2}-2+\left (2 i x^{2}-3 x +i\right ) \sqrt {2}}{2 \sqrt {-2 x^{4}-3 x^{2}+1}}\right )}{2}+\arctan \left (\frac {\sqrt {-2 x^{4}-3 x^{2}+1}}{x}\right )\)	\(125\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 x \sqrt {-2 x^{4}-3 x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {-2 x^{4}-3 x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{4}+2 x^{2}-1}\right )}{2}\)	\(148\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 5 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 4 \, x^{2} - 1}\right ) \]

[In]

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

[Out]

-1/2*sqrt(2)*arctan(2*sqrt(2)*sqrt(-2*x^4 - 3*x^2 + 1)*x/(2*x^4 + 5*x^2 - 1)) + 1/2*arctan(2*sqrt(-2*x^4 - 3*x
^2 + 1)*x/(2*x^4 + 4*x^2 - 1))

Sympy [F]

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

[In]

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

[Out]

Integral((2*x**4 + 1)*sqrt(-2*x**4 - 3*x**2 + 1)/((x**2 + 1)*(2*x**2 - 1)*(2*x**4 + 2*x**2 - 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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