\(\int \frac {\sqrt {2-x^2-4 x^4} (1+2 x^4)}{(-1+2 x^4) (-1-x^2+2 x^4)} \, dx\) [989]

Optimal result

Integrand size = 47, antiderivative size = 75 \[ \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx=\arctan \left (\frac {x \sqrt {2-x^2-4 x^4}}{-2+x^2+4 x^4}\right )-\sqrt {3} \arctan \left (\frac {\sqrt {3} x \sqrt {2-x^2-4 x^4}}{-2+x^2+4 x^4}\right ) \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 1.88 (sec) , antiderivative size = 475, normalized size of antiderivative = 6.33, number of steps used = 32, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6857, 1222, 1194, 538, 435, 430, 1226, 551} \[ \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx=-\frac {1}{4} \sqrt {3 \left (11 \sqrt {33}-59\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {\left (1+4 \sqrt {2}-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+\frac {\left (1-4 \sqrt {2}-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (1-\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (-\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (-1+\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right ) \]

[In]

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

[Out]

((1 - 4*Sqrt[2] - Sqrt[33])*EllipticF[ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16])/Sqrt[2*(1 + S
qrt[33])] + ((1 + 4*Sqrt[2] - Sqrt[33])*EllipticF[ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16])/S
qrt[2*(1 + Sqrt[33])] + (Sqrt[3*(13 + 3*Sqrt[33])]*EllipticF[ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[
33])/16])/4 - (Sqrt[3*(-59 + 11*Sqrt[33])]*EllipticF[ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[33])/16]
)/4 + 3*Sqrt[2/(1 + Sqrt[33])]*EllipticPi[(1 - Sqrt[33])/4, ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x], (-17 + Sqrt[3
3])/16] - Sqrt[2/(1 + Sqrt[33])]*EllipticPi[-1/4*(1 - Sqrt[33])/Sqrt[2], ArcSin[2*Sqrt[2/(-1 + Sqrt[33])]*x],
(-17 + Sqrt[33])/16] - Sqrt[2/(1 + Sqrt[33])]*EllipticPi[(1 - Sqrt[33])/(4*Sqrt[2]), ArcSin[2*Sqrt[2/(-1 + Sqr
t[33])]*x], (-17 + Sqrt[33])/16] + 3*Sqrt[2/(1 + Sqrt[33])]*EllipticPi[(-1 + Sqrt[33])/8, ArcSin[2*Sqrt[2/(-1
+ Sqrt[33])]*x], (-17 + Sqrt[33])/16]

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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {2-x^2-4 x^4}}{-1+x^2}+\frac {2 \sqrt {2-x^2-4 x^4}}{1+2 x^2}-\frac {4 x^2 \sqrt {2-x^2-4 x^4}}{-1+2 x^4}\right ) \, dx \\ & = 2 \int \frac {\sqrt {2-x^2-4 x^4}}{1+2 x^2} \, dx-4 \int \frac {x^2 \sqrt {2-x^2-4 x^4}}{-1+2 x^4} \, dx+\int \frac {\sqrt {2-x^2-4 x^4}}{-1+x^2} \, dx \\ & = -\left (\frac {1}{2} \int \frac {-2+8 x^2}{\sqrt {2-x^2-4 x^4}} \, dx\right )-3 \int \frac {1}{\left (-1+x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx+3 \int \frac {1}{\left (1+2 x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx-4 \int \left (-\frac {\sqrt {2-x^2-4 x^4}}{2 \sqrt {2} \left (1-\sqrt {2} x^2\right )}+\frac {\sqrt {2-x^2-4 x^4}}{2 \sqrt {2} \left (1+\sqrt {2} x^2\right )}\right ) \, dx-\int \frac {5+4 x^2}{\sqrt {2-x^2-4 x^4}} \, dx \\ & = -\left (2 \int \frac {-2+8 x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx\right )-4 \int \frac {5+4 x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx-12 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \left (-1+x^2\right ) \sqrt {1+\sqrt {33}+8 x^2}} \, dx+12 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \left (1+2 x^2\right ) \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\sqrt {2} \int \frac {\sqrt {2-x^2-4 x^4}}{1-\sqrt {2} x^2} \, dx-\sqrt {2} \int \frac {\sqrt {2-x^2-4 x^4}}{1+\sqrt {2} x^2} \, dx \\ & = 3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (1-\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (-1+\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-2 \left (2 \int \frac {\sqrt {1+\sqrt {33}+8 x^2}}{\sqrt {-1+\sqrt {33}-8 x^2}} \, dx\right )-\frac {\int \frac {-4-\sqrt {2}-4 \sqrt {2} x^2}{\sqrt {2-x^2-4 x^4}} \, dx}{\sqrt {2}}+\frac {\int \frac {-4+\sqrt {2}+4 \sqrt {2} x^2}{\sqrt {2-x^2-4 x^4}} \, dx}{\sqrt {2}}-\left (2 \left (9-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\left (2 \left (3+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx-\int \frac {1}{\left (1-\sqrt {2} x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx-\int \frac {1}{\left (1+\sqrt {2} x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx \\ & = -\sqrt {2 \left (1+\sqrt {33}\right )} E\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\frac {1}{4} \sqrt {3 \left (-59+11 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (1-\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (-1+\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-4 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2} \left (1-\sqrt {2} x^2\right )} \, dx-4 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2} \left (1+\sqrt {2} x^2\right )} \, dx-\left (2 \sqrt {2}\right ) \int \frac {-4-\sqrt {2}-4 \sqrt {2} x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\left (2 \sqrt {2}\right ) \int \frac {-4+\sqrt {2}+4 \sqrt {2} x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx \\ & = -\sqrt {2 \left (1+\sqrt {33}\right )} E\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\frac {1}{4} \sqrt {3 \left (-59+11 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (1-\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (-\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (-1+\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+2 \left (2 \int \frac {\sqrt {1+\sqrt {33}+8 x^2}}{\sqrt {-1+\sqrt {33}-8 x^2}} \, dx\right )+\left (2 \left (1-4 \sqrt {2}-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\left (2 \left (1+4 \sqrt {2}-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx \\ & = \frac {\left (1-4 \sqrt {2}-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+\frac {\left (1+4 \sqrt {2}-\sqrt {33}\right ) \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\frac {1}{4} \sqrt {3 \left (-59+11 \sqrt {33}\right )} \operatorname {EllipticF}\left (\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (1-\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (-\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1-\sqrt {33}}{4 \sqrt {2}},\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \operatorname {EllipticPi}\left (\frac {1}{8} \left (-1+\sqrt {33}\right ),\arcsin \left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right ),\frac {1}{16} \left (-17+\sqrt {33}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {2-x^2-4 x^4}}\right )+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{\sqrt {2-x^2-4 x^4}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 18.46 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81

method	result	size
elliptic	\(\frac {\left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-4 x^{4}-x^{2}+2}}{x}\right )-\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {-4 x^{4}-x^{2}+2}\, \sqrt {2}}{6 x}\right )\right ) \sqrt {2}}{2}\)	\(61\)
default	\(\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (i \left (x -1\right )^{2} \sqrt {2}+2 x^{2}+\frac {x}{2}-1\right ) \sqrt {3}}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (i \left (1+x \right )^{2} \sqrt {2}+2 x^{2}-\frac {x}{2}-1\right ) \sqrt {3}}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}+\arctan \left (\frac {\sqrt {-4 x^{4}-x^{2}+2}}{x}\right )\)	\(113\)
pseudoelliptic	\(\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (i \left (x -1\right )^{2} \sqrt {2}+2 x^{2}+\frac {x}{2}-1\right ) \sqrt {3}}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (i \left (1+x \right )^{2} \sqrt {2}+2 x^{2}-\frac {x}{2}-1\right ) \sqrt {3}}{3 \sqrt {-4 x^{4}-x^{2}+2}}\right )}{2}+\arctan \left (\frac {\sqrt {-4 x^{4}-x^{2}+2}}{x}\right )\)	\(113\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+3 x \sqrt {-4 x^{4}-x^{2}+2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\left (x -1\right ) \left (1+x \right ) \left (2 x^{2}+1\right )}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+x \sqrt {-4 x^{4}-x^{2}+2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{4}-1}\right )}{2}\)	\(143\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-4 \, x^{4} - x^{2} + 2} x}{2 \, x^{4} + 2 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-4 \, x^{4} - x^{2} + 2} x}{2 \, x^{4} + x^{2} - 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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