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

Optimal. Leaf size=75 \[ \tan ^{-1}\left (\frac {x \sqrt {-4 x^4-x^2+2}}{4 x^4+x^2-2}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-4 x^4-x^2+2}}{4 x^4+x^2-2}\right ) \]

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Rubi [C]  time = 1.79, antiderivative size = 475, normalized size of antiderivative = 6.33, number of steps used = 32, number of rules used = 8, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6725, 1208, 1180, 524, 424, 419, 1212, 537} \begin {gather*} -\frac {1}{4} \sqrt {3 \left (11 \sqrt {33}-59\right )} F\left (\sin ^{-1}\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 )} F\left (\sin ^{-1}\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 ) F\left (\sin ^{-1}\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 ) F\left (\sin ^{-1}\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}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\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}}} \Pi \left (-\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\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}}} \Pi \left (\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\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}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[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 419

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

Rule 424

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

Rule 524

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 1180

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 1208

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 1212

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 6725

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*} \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 \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}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\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}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\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 (\sin ^{-1}\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 )} F\left (\sin ^{-1}\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 )} F\left (\sin ^{-1}\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}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\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}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\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 (\sin ^{-1}\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 )} F\left (\sin ^{-1}\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 )} F\left (\sin ^{-1}\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}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\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}}} \Pi \left (-\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\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}}} \Pi \left (\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\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}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\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 ) F\left (\sin ^{-1}\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 ) F\left (\sin ^{-1}\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 )} F\left (\sin ^{-1}\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 )} F\left (\sin ^{-1}\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}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\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}}} \Pi \left (-\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\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}}} \Pi \left (\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\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}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )\\ \end {align*}

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Mathematica [C]  time = 7.04, size = 2667, normalized size = 35.56 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[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]

-(((-1 + x)*(1 + x)*(1 + 2*x^2)*Sqrt[2 - x^2 - 4*x^4]*(-1 + 2*x^4)*(Sqrt[2 - x^2 - 4*x^4]/(2*(-1 + x)) - Sqrt[
2 - x^2 - 4*x^4]/(2*(1 + x)) + (2*Sqrt[2 - x^2 - 4*x^4])/(1 + 2*x^2) - (4*x^2*Sqrt[2 - x^2 - 4*x^4])/(-1 + 2*x
^4))*((I*Sqrt[(1 + Sqrt[33])/2]*Sqrt[1 - (8*x^2)/(-1 - Sqrt[33])]*Sqrt[1 - (8*x^2)/(-1 + Sqrt[33])]*EllipticF[
I*ArcSinh[2*Sqrt[2/(1 + Sqrt[33])]*x], (-1 - Sqrt[33])/(-1 + Sqrt[33])])/Sqrt[2 - x^2 - 4*x^4] + ((I/8)*Sqrt[1
+ Sqrt[33]]*Sqrt[1 - (8*x^2)/(-1 - Sqrt[33])]*Sqrt[1 - (8*x^2)/(-1 + Sqrt[33])]*EllipticPi[-1/4*(-1 - Sqrt[33
])/Sqrt[2], I*ArcSinh[2*Sqrt[2/(1 + Sqrt[33])]*x], (-1 - Sqrt[33])/(-1 + Sqrt[33])])/((-1 - 1/Sqrt[2])*(1/2 -
1/Sqrt[2])*Sqrt[2 - x^2 - 4*x^4]) - ((I/8)*Sqrt[1 + Sqrt[33]]*Sqrt[1 - (8*x^2)/(-1 - Sqrt[33])]*Sqrt[1 - (8*x^
2)/(-1 + Sqrt[33])]*EllipticPi[(-1 - Sqrt[33])/(4*Sqrt[2]), I*ArcSinh[2*Sqrt[2/(1 + Sqrt[33])]*x], (-1 - Sqrt[
33])/(-1 + Sqrt[33])])/((-1 + 1/Sqrt[2])*(1/2 + 1/Sqrt[2])*Sqrt[2 - x^2 - 4*x^4]) + (((3*I)/8)*Sqrt[(1 + Sqrt[
33])/2]*Sqrt[1 - (8*x^2)/(-1 - Sqrt[33])]*Sqrt[1 - (8*x^2)/(-1 + Sqrt[33])]*EllipticPi[(1 + Sqrt[33])/4, I*Arc
Sinh[2*Sqrt[2/(1 + Sqrt[33])]*x], (-1 - Sqrt[33])/(-1 + Sqrt[33])])/((-1/2 - 1/Sqrt[2])*(-1/2 + 1/Sqrt[2])*Sqr
t[2 - x^2 - 4*x^4]) - (3*(((Sqrt[(-1 + Sqrt[33])/2]/2 + (I/2)*Sqrt[(1 + Sqrt[33])/2])*(-1/2*Sqrt[(-1 + Sqrt[33
])/2] + x)^2*Sqrt[((-1/2*I)*Sqrt[(1 + Sqrt[33])/2] + x)/((Sqrt[(-1 + Sqrt[33])/2]/2 + (I/2)*Sqrt[(1 + Sqrt[33]
)/2])*(-1/2*Sqrt[(-1 + Sqrt[33])/2] + x))]*Sqrt[((I/2)*Sqrt[(1 + Sqrt[33])/2] + x)/((Sqrt[(-1 + Sqrt[33])/2]/2
- (I/2)*Sqrt[(1 + Sqrt[33])/2])*(-1/2*Sqrt[(-1 + Sqrt[33])/2] + x))]*Sqrt[((Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 +
Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] + 4*x))/((Sqrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[3
3])] - 4*x))]*((1 + Sqrt[(-1 + Sqrt[33])/2]/2)*EllipticF[ArcSin[Sqrt[((Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[3
3]])*(Sqrt[2*(-1 + Sqrt[33])] + 4*x))/((Sqrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] -
4*x))]], (Sqrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]])^2/(Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[33]])^2] - Sqr
t[(-1 + Sqrt[33])/2]*EllipticPi[((-1 + Sqrt[(-1 + Sqrt[33])/2]/2)*(Sqrt[(-1 + Sqrt[33])/2]/2 + (I/2)*Sqrt[(1 +
Sqrt[33])/2]))/((-1 - Sqrt[(-1 + Sqrt[33])/2]/2)*(-1/2*Sqrt[(-1 + Sqrt[33])/2] + (I/2)*Sqrt[(1 + Sqrt[33])/2]
)), ArcSin[Sqrt[((Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] + 4*x))/((Sqrt[-1 + Sqr
t[33]] + I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] - 4*x))]], (Sqrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]]
)^2/(Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[33]])^2]))/((-1 - Sqrt[(-1 + Sqrt[33])/2]/2)*(1 - Sqrt[(-1 + Sqrt[3
3])/2]/2)*(Sqrt[(-1 + Sqrt[33])/2]/2 - (I/2)*Sqrt[(1 + Sqrt[33])/2])*Sqrt[2 - x^2 - 4*x^4]) - ((Sqrt[(-1 + Sqr
t[33])/2]/2 + (I/2)*Sqrt[(1 + Sqrt[33])/2])*(-1/2*Sqrt[(-1 + Sqrt[33])/2] + x)^2*Sqrt[((-1/2*I)*Sqrt[(1 + Sqrt
[33])/2] + x)/((Sqrt[(-1 + Sqrt[33])/2]/2 + (I/2)*Sqrt[(1 + Sqrt[33])/2])*(-1/2*Sqrt[(-1 + Sqrt[33])/2] + x))]
*Sqrt[((I/2)*Sqrt[(1 + Sqrt[33])/2] + x)/((Sqrt[(-1 + Sqrt[33])/2]/2 - (I/2)*Sqrt[(1 + Sqrt[33])/2])*(-1/2*Sqr
t[(-1 + Sqrt[33])/2] + x))]*Sqrt[((Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] + 4*x)
)/((Sqrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] - 4*x))]*((-1 + Sqrt[(-1 + Sqrt[33])/
2]/2)*EllipticF[ArcSin[Sqrt[((Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] + 4*x))/((S
qrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] - 4*x))]], (Sqrt[-1 + Sqrt[33]] + I*Sqrt[1
+ Sqrt[33]])^2/(Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[33]])^2] - Sqrt[(-1 + Sqrt[33])/2]*EllipticPi[((1 + Sqr
t[(-1 + Sqrt[33])/2]/2)*(Sqrt[(-1 + Sqrt[33])/2]/2 + (I/2)*Sqrt[(1 + Sqrt[33])/2]))/((1 - Sqrt[(-1 + Sqrt[33])
/2]/2)*(-1/2*Sqrt[(-1 + Sqrt[33])/2] + (I/2)*Sqrt[(1 + Sqrt[33])/2])), ArcSin[Sqrt[((Sqrt[-1 + Sqrt[33]] - I*S
qrt[1 + Sqrt[33]])*(Sqrt[2*(-1 + Sqrt[33])] + 4*x))/((Sqrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]])*(Sqrt[2*(-1
+ Sqrt[33])] - 4*x))]], (Sqrt[-1 + Sqrt[33]] + I*Sqrt[1 + Sqrt[33]])^2/(Sqrt[-1 + Sqrt[33]] - I*Sqrt[1 + Sqrt[
33]])^2]))/((-1 - Sqrt[(-1 + Sqrt[33])/2]/2)*(1 - Sqrt[(-1 + Sqrt[33])/2]/2)*(Sqrt[(-1 + Sqrt[33])/2]/2 - (I/2
)*Sqrt[(1 + Sqrt[33])/2])*Sqrt[2 - x^2 - 4*x^4])))/(2*(1 - 1/Sqrt[2])*(1 + 1/Sqrt[2]))))/((1 + 2*x^4)*(-2 + x^
2 + 4*x^4)))

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IntegrateAlgebraic [A]  time = 0.50, size = 75, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x \sqrt {2-x^2-4 x^4}}{-2+x^2+4 x^4}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {2-x^2-4 x^4}}{-2+x^2+4 x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 0.54, size = 71, normalized size = 0.95 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.68, size = 61, normalized size = 0.81

method	result	size
elliptic	\(\frac {\left (-\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {-4 x^{4}-x^{2}+2}\, \sqrt {2}}{6 x}\right )+\sqrt {2}\, \arctan \left (\frac {\sqrt {-4 x^{4}-x^{2}+2}}{x}\right )\right ) \sqrt {2}}{2}\)	\(61\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+3 \sqrt {-4 x^{4}-x^{2}+2}\, x -\RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (-1+x \right ) \left (1+x \right ) \left (2 x^{2}+1\right )}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\sqrt {-4 x^{4}-x^{2}+2}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{2 x^{4}-1}\right )}{2}\)	\(145\)
default	\(-\frac {6 \sqrt {1-\left (\frac {1}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {1+\sqrt {33}}}{2}, \frac {i \sqrt {66}}{8}-\frac {i \sqrt {2}}{8}\right )}{\sqrt {1+\sqrt {33}}\, \sqrt {-4 x^{4}-x^{2}+2}}-\frac {32 \sqrt {1-\left (\frac {1}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {1+\sqrt {33}}}{2}, \frac {i \sqrt {66}}{8}-\frac {i \sqrt {2}}{8}\right )-\EllipticE \left (\frac {x \sqrt {1+\sqrt {33}}}{2}, \frac {i \sqrt {66}}{8}-\frac {i \sqrt {2}}{8}\right )\right )}{\sqrt {1+\sqrt {33}}\, \sqrt {-4 x^{4}-x^{2}+2}\, \left (-1+\sqrt {33}\right )}+\frac {3 \sqrt {1-\left (\frac {1}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}\, x , \frac {1}{\frac {1}{4}+\frac {\sqrt {33}}{4}}, \frac {\sqrt {\frac {1}{4}-\frac {\sqrt {33}}{4}}}{\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}}\right )}{\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}\, \sqrt {-4 x^{4}-x^{2}+2}}+\frac {2 \sqrt {1-\frac {x^{2}}{4}-\frac {x^{2} \sqrt {33}}{4}}\, \sqrt {1-\frac {x^{2}}{4}+\frac {x^{2} \sqrt {33}}{4}}\, \EllipticF \left (\frac {x \sqrt {1+\sqrt {33}}}{2}, \frac {i \sqrt {66}}{8}-\frac {i \sqrt {2}}{8}\right )}{\sqrt {1+\sqrt {33}}\, \sqrt {-4 x^{4}-x^{2}+2}}+\frac {32 \sqrt {1-\frac {x^{2}}{4}-\frac {x^{2} \sqrt {33}}{4}}\, \sqrt {1-\frac {x^{2}}{4}+\frac {x^{2} \sqrt {33}}{4}}\, \EllipticF \left (\frac {x \sqrt {1+\sqrt {33}}}{2}, \frac {i \sqrt {66}}{8}-\frac {i \sqrt {2}}{8}\right )}{\sqrt {1+\sqrt {33}}\, \sqrt {-4 x^{4}-x^{2}+2}\, \left (-1+\sqrt {33}\right )}-\frac {32 \sqrt {1-\frac {x^{2}}{4}-\frac {x^{2} \sqrt {33}}{4}}\, \sqrt {1-\frac {x^{2}}{4}+\frac {x^{2} \sqrt {33}}{4}}\, \EllipticE \left (\frac {x \sqrt {1+\sqrt {33}}}{2}, \frac {i \sqrt {66}}{8}-\frac {i \sqrt {2}}{8}\right )}{\sqrt {1+\sqrt {33}}\, \sqrt {-4 x^{4}-x^{2}+2}\, \left (-1+\sqrt {33}\right )}+\frac {3 \sqrt {1-\frac {x^{2}}{4}-\frac {x^{2} \sqrt {33}}{4}}\, \sqrt {1-\frac {x^{2}}{4}+\frac {x^{2} \sqrt {33}}{4}}\, \EllipticPi \left (\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}\, x , -\frac {2}{\frac {1}{4}+\frac {\sqrt {33}}{4}}, \frac {\sqrt {\frac {1}{4}-\frac {\sqrt {33}}{4}}}{\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}}\right )}{\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}\, \sqrt {-4 x^{4}-x^{2}+2}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (\frac {\arctanh \left (\frac {\left (8 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-33 \underline {\hspace {1.25 ex}}\alpha ^{2}+31 x^{2}+8\right )}{62 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {-4 x^{4}-x^{2}+2}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {\sqrt {4}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {-x^{2}+4-x^{2} \sqrt {33}}\, \sqrt {-x^{2}+4+x^{2} \sqrt {33}}\, \EllipticPi \left (\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {33}}{4}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{4}, \frac {\sqrt {\frac {1}{4}-\frac {\sqrt {33}}{4}}}{\sqrt {\frac {1}{4}+\frac {\sqrt {33}}{4}}}\right )}{\sqrt {1+\sqrt {33}}\, \sqrt {-4 x^{4}-x^{2}+2}}\right )\right )}{4}\)	\(823\)

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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