3.18.89 \(\int \frac {(2+x^2) (-2-2 x+x^2) \sqrt {4-3 x^2+x^4}}{x^2 (-2+x^2) (-4+x+2 x^2)} \, dx\)

Optimal. Leaf size=121 \[ \frac {\sqrt {x^4-3 x^2+4}}{2 x}-\frac {5}{4} \log \left (x^2+\sqrt {x^4-3 x^2+4}-2\right )+4 \tanh ^{-1}\left (\frac {x}{x^2+\sqrt {x^4-3 x^2+4}-2}\right )-\frac {5}{2} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 x^2+2 \sqrt {x^4-3 x^2+4}+x-4}\right )+\frac {5 \log (x)}{4} \]

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Rubi [C]  time = 3.65, antiderivative size = 917, normalized size of antiderivative = 7.58, number of steps used = 53, number of rules used = 21, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6725, 1117, 1197, 1103, 1195, 1114, 734, 843, 619, 215, 724, 206, 1208, 1210, 1698, 207, 6728, 1728, 1216, 1706, 1247} \begin {gather*} \frac {5}{64} \left (7+\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7-\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-3 x^2+4}}\right )-\frac {5}{4} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {x^4-3 x^2+4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {2 \left (5-\sqrt {33}\right ) x^2+3 \sqrt {33}+13}{2 \sqrt {10 \left (17-\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {2 \left (5+\sqrt {33}\right ) x^2-3 \sqrt {33}+13}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )-\frac {25 \left (17+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {x^4-3 x^2+4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {x^4-3 x^2+4}}-\frac {5 \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33-17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {\sqrt {x^4-3 x^2+4}}{2 x}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}-\frac {5}{16} \sqrt {x^4-3 x^2+4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-5*Sqrt[4 - 3*x^2 + x^4])/16 + (5*(1 - Sqrt[33])*Sqrt[4 - 3*x^2 + x^4])/32 + (5*(1 + Sqrt[33])*Sqrt[4 - 3*x^2
+ x^4])/32 + Sqrt[4 - 3*x^2 + x^4]/(2*x) - (15*ArcSinh[(3 - 2*x^2)/Sqrt[7]])/32 + (5*(7 - Sqrt[33])*ArcSinh[(
3 - 2*x^2)/Sqrt[7]])/64 + (5*(7 + Sqrt[33])*ArcSinh[(3 - 2*x^2)/Sqrt[7]])/64 + 2*ArcTanh[x/Sqrt[4 - 3*x^2 + x^
4]] - (5*Sqrt[5]*ArcTanh[(Sqrt[5]*x)/(2*Sqrt[4 - 3*x^2 + x^4])])/4 + (5*ArcTanh[(8 - 3*x^2)/(4*Sqrt[4 - 3*x^2
+ x^4])])/8 - (5*Sqrt[5]*ArcTanh[(13 + 3*Sqrt[33] + 2*(5 - Sqrt[33])*x^2)/(2*Sqrt[10*(17 - Sqrt[33])]*Sqrt[4 -
3*x^2 + x^4])])/8 + (5*Sqrt[5]*ArcTanh[(13 - 3*Sqrt[33] + 2*(5 + Sqrt[33])*x^2)/(2*Sqrt[10*(17 + Sqrt[33])]*S
qrt[4 - 3*x^2 + x^4])])/8 - (5*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/
8])/(4*Sqrt[2]*Sqrt[4 - 3*x^2 + x^4]) + (5*(9 - Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*Ellipt
icF[2*ArcTan[x/Sqrt[2]], 7/8])/(32*Sqrt[2]*Sqrt[4 - 3*x^2 + x^4]) - (25*(17 - Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*
x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/8])/(16*Sqrt[2]*(33 - Sqrt[33])*Sqrt[4 - 3*x^2 + x^4]
) + (5*(9 + Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 7/8])/(32*S
qrt[2]*Sqrt[4 - 3*x^2 + x^4]) - (25*(17 + Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*
ArcTan[x/Sqrt[2]], 7/8])/(16*Sqrt[2]*(33 + Sqrt[33])*Sqrt[4 - 3*x^2 + x^4]) + (25*(17 - Sqrt[33])*(2 + x^2)*Sq
rt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[33/32, 2*ArcTan[x/Sqrt[2]], 7/8])/(32*Sqrt[2]*(33 - 17*Sqrt[33])*
Sqrt[4 - 3*x^2 + x^4]) + (25*(17 + Sqrt[33])*(2 + x^2)*Sqrt[(4 - 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[33/32, 2
*ArcTan[x/Sqrt[2]], 7/8])/(32*Sqrt[2]*(33 + 17*Sqrt[33])*Sqrt[4 - 3*x^2 + x^4])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 215

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

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 724

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

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

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

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 1117

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2
+ c*x^4)^p)/(d*(m + 1)), x] - Dist[(2*p)/(d^2*(m + 1)), Int[(d*x)^(m + 2)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p
- 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && LtQ[m, -1] && IntegerQ[2*p] &&
(IntegerQ[p] || IntegerQ[m])

Rule 1195

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

Rule 1197

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

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 1210

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

Rule 1216

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

Rule 1247

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

Rule 1698

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B},
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 1706

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[
{q = Rt[B/A, 2]}, -Simp[((B*d - A*e)*ArcTan[(Rt[-b + (c*d)/e + (a*e)/d, 2]*x)/Sqrt[a + b*x^2 + c*x^4]])/(2*d*e
*Rt[-b + (c*d)/e + (a*e)/d, 2]), x] + Simp[((B*d + A*e)*(A + B*x^2)*Sqrt[(A^2*(a + b*x^2 + c*x^4))/(a*(A + B*x
^2)^2)]*EllipticPi[Cancel[-((B*d - A*e)^2/(4*d*e*A*B))], 2*ArcTan[q*x], 1/2 - (b*A)/(4*a*B)])/(4*d*e*A*q*Sqrt[
a + b*x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^
2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1728

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

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]

Rule 6728

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*} \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx &=\int \left (-\frac {\sqrt {4-3 x^2+x^4}}{2 x^2}-\frac {5 \sqrt {4-3 x^2+x^4}}{8 x}-\frac {4 \sqrt {4-3 x^2+x^4}}{-2+x^2}+\frac {5 (17+2 x) \sqrt {4-3 x^2+x^4}}{8 \left (-4+x+2 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {4-3 x^2+x^4}}{x^2} \, dx\right )-\frac {5}{8} \int \frac {\sqrt {4-3 x^2+x^4}}{x} \, dx+\frac {5}{8} \int \frac {(17+2 x) \sqrt {4-3 x^2+x^4}}{-4+x+2 x^2} \, dx-4 \int \frac {\sqrt {4-3 x^2+x^4}}{-2+x^2} \, dx\\ &=\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5}{16} \operatorname {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{x} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {-3+2 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{8} \int \left (\frac {\left (2+2 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}{1-\sqrt {33}+4 x}+\frac {\left (2-2 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}{1+\sqrt {33}+4 x}\right ) \, dx+4 \int \frac {1-x^2}{\sqrt {4-3 x^2+x^4}} \, dx-8 \int \frac {1}{\left (-2+x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}+\frac {5}{32} \operatorname {Subst}\left (\int \frac {-8+3 x}{x \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {-2-x^2}{\left (-2+x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx-4 \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+8 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {1}{4} \left (5 \left (1-\sqrt {33}\right )\right ) \int \frac {\sqrt {4-3 x^2+x^4}}{1+\sqrt {33}+4 x} \, dx+\frac {1}{4} \left (5 \left (1+\sqrt {33}\right )\right ) \int \frac {\sqrt {4-3 x^2+x^4}}{1-\sqrt {33}+4 x} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {15}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-4 \operatorname {Subst}\left (\int \frac {1}{-2+2 x^2} \, dx,x,\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-40 \int \frac {\sqrt {4-3 x^2+x^4}}{\left (1-\sqrt {33}\right )^2-16 x^2} \, dx-40 \int \frac {\sqrt {4-3 x^2+x^4}}{\left (1+\sqrt {33}\right )^2-16 x^2} \, dx-\left (5 \left (1-\sqrt {33}\right )\right ) \int \frac {x \sqrt {4-3 x^2+x^4}}{\left (1+\sqrt {33}\right )^2-16 x^2} \, dx-\left (5 \left (1+\sqrt {33}\right )\right ) \int \frac {x \sqrt {4-3 x^2+x^4}}{\left (1-\sqrt {33}\right )^2-16 x^2} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5}{32} \int \frac {-48+\left (1-\sqrt {33}\right )^2+16 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{32} \int \frac {-48+\left (1+\sqrt {33}\right )^2+16 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {8-3 x^2}{\sqrt {4-3 x^2+x^4}}\right )+\frac {15 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,-3+2 x^2\right )}{32 \sqrt {7}}-\frac {1}{2} \left (5 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{\left (1+\sqrt {33}\right )^2-16 x} \, dx,x,x^2\right )-\frac {1}{4} \left (25 \left (17-\sqrt {33}\right )\right ) \int \frac {1}{\left (\left (1-\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx-\frac {1}{2} \left (5 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{\left (1-\sqrt {33}\right )^2-16 x} \, dx,x,x^2\right )-\frac {1}{4} \left (25 \left (17+\sqrt {33}\right )\right ) \int \frac {1}{\left (\left (1+\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (5 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx\right )-\frac {1}{64} \left (5 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {2 \left (13-3 \sqrt {33}\right )+4 \left (5+\sqrt {33}\right ) x}{\left (\left (1+\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (5 \left (9-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx-\frac {1}{64} \left (5 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {2 \left (13+3 \sqrt {33}\right )+4 \left (5-\sqrt {33}\right ) x}{\left (\left (1-\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (5 \left (9+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx-\frac {\left (25 \left (17+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx}{8 \left (33+\sqrt {33}\right )}-\frac {\left (100 \left (17+\sqrt {33}\right )\right ) \int \frac {1+\frac {x^2}{2}}{\left (\left (1+\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx}{33+\sqrt {33}}-\frac {\left (400 \left (17-\sqrt {33}\right ) \left (-16+\frac {1}{2} \left (1-\sqrt {33}\right )^2\right )\right ) \int \frac {1+\frac {x^2}{2}}{\left (\left (1-\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx}{-1024+\left (1-\sqrt {33}\right )^4}-\frac {\left (25 \left (17-\sqrt {33}\right ) \left (-32+\left (1-\sqrt {33}\right )^2\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx}{4 \left (-1024+\left (1-\sqrt {33}\right )^4\right )}\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {165} \left (17-\sqrt {33}\right ) \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )}{8 \left (33-17 \sqrt {33}\right )}+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (-\frac {5 x \sqrt {4-3 x^2+x^4}}{2 \left (2+x^2\right )}+\frac {5 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {2} \sqrt {4-3 x^2+x^4}}\right )+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \left (33 \sqrt {2}-17 \sqrt {66}\right ) \sqrt {4-3 x^2+x^4}}+\frac {1}{4} \left (25 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{64} \left (5 \left (7-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (25 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{64} \left (5 \left (7+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {165} \left (17-\sqrt {33}\right ) \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )}{8 \left (33-17 \sqrt {33}\right )}+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (-\frac {5 x \sqrt {4-3 x^2+x^4}}{2 \left (2+x^2\right )}+\frac {5 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {2} \sqrt {4-3 x^2+x^4}}\right )+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \left (33 \sqrt {2}-17 \sqrt {66}\right ) \sqrt {4-3 x^2+x^4}}-\frac {1}{2} \left (25 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4096-192 \left (1-\sqrt {33}\right )^2+4 \left (1-\sqrt {33}\right )^4-x^2} \, dx,x,\frac {-128+3 \left (1-\sqrt {33}\right )^2-4 \left (5-\sqrt {33}\right ) x^2}{\sqrt {4-3 x^2+x^4}}\right )-\frac {\left (5 \left (7-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,-3+2 x^2\right )}{64 \sqrt {7}}-\frac {1}{2} \left (25 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4096-192 \left (1+\sqrt {33}\right )^2+4 \left (1+\sqrt {33}\right )^4-x^2} \, dx,x,\frac {-128+3 \left (1+\sqrt {33}\right )^2-4 \left (5+\sqrt {33}\right ) x^2}{\sqrt {4-3 x^2+x^4}}\right )-\frac {\left (5 \left (7+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,-3+2 x^2\right )}{64 \sqrt {7}}\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7-\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7+\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {165} \left (17-\sqrt {33}\right ) \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )}{8 \left (33-17 \sqrt {33}\right )}+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {13+3 \sqrt {33}+2 \left (5-\sqrt {33}\right ) x^2}{2 \sqrt {10 \left (17-\sqrt {33}\right )} \sqrt {4-3 x^2+x^4}}\right )+\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {13-3 \sqrt {33}+2 \left (5+\sqrt {33}\right ) x^2}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (-\frac {5 x \sqrt {4-3 x^2+x^4}}{2 \left (2+x^2\right )}+\frac {5 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {2} \sqrt {4-3 x^2+x^4}}\right )+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \left (33 \sqrt {2}-17 \sqrt {66}\right ) \sqrt {4-3 x^2+x^4}}\\ \end {align*}

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Mathematica [C]  time = 1.60, size = 871, normalized size = 7.20 \begin {gather*} \frac {8 \sqrt {-\frac {i}{3 i+\sqrt {7}}} x^4-24 \sqrt {-\frac {i}{3 i+\sqrt {7}}} x^2+10 \sqrt {-\frac {i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right ) x+10 \sqrt {-\frac {i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {x^4-3 x^2+4}}\right ) x+10 \sqrt {-\frac {5 i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \tanh ^{-1}\left (\frac {2 \left (5+\sqrt {33}\right ) x^2-3 \sqrt {33}+13}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right ) x-10 \sqrt {-\frac {5 i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \tanh ^{-1}\left (\frac {-2 \left (-5+\sqrt {33}\right ) x^2+3 \sqrt {33}+13}{2 \sqrt {10} \sqrt {-\left (\left (-17+\sqrt {33}\right ) \left (x^4-3 x^2+4\right )\right )}}\right ) x-9 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x-32 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} \Pi \left (\frac {3}{4}-\frac {i \sqrt {7}}{4};i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x+25 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} \Pi \left (\frac {4 i \left (3 i+\sqrt {7}\right )}{-17+\sqrt {33}};i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x+25 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} \Pi \left (\frac {12-4 i \sqrt {7}}{17+\sqrt {33}};i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x+32 \sqrt {-\frac {i}{3 i+\sqrt {7}}}}{16 \sqrt {-\frac {i}{3 i+\sqrt {7}}} x \sqrt {x^4-3 x^2+4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(32*Sqrt[(-I)/(3*I + Sqrt[7])] - 24*Sqrt[(-I)/(3*I + Sqrt[7])]*x^2 + 8*Sqrt[(-I)/(3*I + Sqrt[7])]*x^4 + 10*Sqr
t[(-I)/(3*I + Sqrt[7])]*x*Sqrt[4 - 3*x^2 + x^4]*ArcSinh[(3 - 2*x^2)/Sqrt[7]] + 10*Sqrt[(-I)/(3*I + Sqrt[7])]*x
*Sqrt[4 - 3*x^2 + x^4]*ArcTanh[(8 - 3*x^2)/(4*Sqrt[4 - 3*x^2 + x^4])] + 10*Sqrt[(-5*I)/(3*I + Sqrt[7])]*x*Sqrt
[4 - 3*x^2 + x^4]*ArcTanh[(13 - 3*Sqrt[33] + 2*(5 + Sqrt[33])*x^2)/(2*Sqrt[10*(17 + Sqrt[33])]*Sqrt[4 - 3*x^2
+ x^4])] - 10*Sqrt[(-5*I)/(3*I + Sqrt[7])]*x*Sqrt[4 - 3*x^2 + x^4]*ArcTanh[(13 + 3*Sqrt[33] - 2*(-5 + Sqrt[33]
)*x^2)/(2*Sqrt[10]*Sqrt[-((-17 + Sqrt[33])*(4 - 3*x^2 + x^4))])] - (9*I)*x*Sqrt[1 + ((2*I)*x^2)/(-3*I + Sqrt[7
])]*Sqrt[2 - ((4*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(3*I + Sqrt[7])]*x], (3*I + Sqrt[7])
/(3*I - Sqrt[7])] - (32*I)*x*Sqrt[1 + ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[2 - ((4*I)*x^2)/(3*I + Sqrt[7])]*Elli
pticPi[3/4 - (I/4)*Sqrt[7], I*ArcSinh[Sqrt[(-2*I)/(3*I + Sqrt[7])]*x], (3*I + Sqrt[7])/(3*I - Sqrt[7])] + (25*
I)*x*Sqrt[1 + ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[2 - ((4*I)*x^2)/(3*I + Sqrt[7])]*EllipticPi[((4*I)*(3*I + Sqr
t[7]))/(-17 + Sqrt[33]), I*ArcSinh[Sqrt[(-2*I)/(3*I + Sqrt[7])]*x], (3*I + Sqrt[7])/(3*I - Sqrt[7])] + (25*I)*
x*Sqrt[1 + ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[2 - ((4*I)*x^2)/(3*I + Sqrt[7])]*EllipticPi[(12 - (4*I)*Sqrt[7])
/(17 + Sqrt[33]), I*ArcSinh[Sqrt[(-2*I)/(3*I + Sqrt[7])]*x], (3*I + Sqrt[7])/(3*I - Sqrt[7])])/(16*Sqrt[(-I)/(
3*I + Sqrt[7])]*x*Sqrt[4 - 3*x^2 + x^4])

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IntegrateAlgebraic [A]  time = 1.24, size = 121, normalized size = 1.00 \begin {gather*} \frac {\sqrt {4-3 x^2+x^4}}{2 x}+4 \tanh ^{-1}\left (\frac {x}{-2+x^2+\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{2} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{-4+x+2 x^2+2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \log (x)}{4}-\frac {5}{4} \log \left (-2+x^2+\sqrt {4-3 x^2+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[4 - 3*x^2 + x^4]/(2*x) + 4*ArcTanh[x/(-2 + x^2 + Sqrt[4 - 3*x^2 + x^4])] - (5*Sqrt[5]*ArcTanh[(Sqrt[5]*x)
/(-4 + x + 2*x^2 + 2*Sqrt[4 - 3*x^2 + x^4])])/2 + (5*Log[x])/4 - (5*Log[-2 + x^2 + Sqrt[4 - 3*x^2 + x^4]])/4

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fricas [A]  time = 0.76, size = 150, normalized size = 1.24 \begin {gather*} \frac {5 \, \sqrt {5} x \log \left (\frac {6 \, x^{4} - 4 \, x^{3} + 2 \, \sqrt {5} \sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} - 15 \, x^{2} + 8 \, x + 24}{4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 8 \, x + 16}\right ) + 16 \, x \log \left (-\frac {x + \sqrt {x^{4} - 3 \, x^{2} + 4}}{x^{2} - 2}\right ) + 10 \, x \log \left (-\frac {x^{2} - \sqrt {x^{4} - 3 \, x^{2} + 4} - 2}{x}\right ) + 4 \, \sqrt {x^{4} - 3 \, x^{2} + 4}}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(5*sqrt(5)*x*log((6*x^4 - 4*x^3 + 2*sqrt(5)*sqrt(x^4 - 3*x^2 + 4)*(x^2 - 2*x - 2) - 15*x^2 + 8*x + 24)/(4*
x^4 + 4*x^3 - 15*x^2 - 8*x + 16)) + 16*x*log(-(x + sqrt(x^4 - 3*x^2 + 4))/(x^2 - 2)) + 10*x*log(-(x^2 - sqrt(x
^4 - 3*x^2 + 4) - 2)/x) + 4*sqrt(x^4 - 3*x^2 + 4))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 2.13, size = 360, normalized size = 2.98

method	result	size
elliptic	\(-\frac {5 \arcsinh \left (\frac {2 \sqrt {7}\, \left (x^{2}-\frac {3}{2}\right )}{7}\right )}{8}-\frac {160 \arctanh \left (\frac {-3 x^{2}+8}{4 \sqrt {x^{4}-3 x^{2}+4}}\right )}{\left (17+\sqrt {33}\right ) \left (-17+\sqrt {33}\right )}-\frac {25 \left (-297+25 \sqrt {33}\right ) \sqrt {33}\, \arctanh \left (\frac {\frac {85}{4}-\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )+170-10 \sqrt {33}}}\right )}{1056 \left (-17+\sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right )}+\frac {25 \left (297+25 \sqrt {33}\right ) \sqrt {33}\, \arctanh \left (\frac {\frac {85}{4}+\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )+170+10 \sqrt {33}}}\right )}{1056 \left (17+\sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right )}+\frac {\left (\frac {\sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{2 x}-\frac {5 \sqrt {10}\, \arctanh \left (\frac {\sqrt {10}\, \sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{5 x}\right )}{4}+2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}-3 x^{2}+4}}{x}\right )\right ) \sqrt {2}}{2}\)	\(360\)
trager	\(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+\frac {5 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-5\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-5\right ) x +5 \sqrt {x^{4}-3 x^{2}+4}-2 \RootOf \left (\textit {\_Z}^{2}-5\right )}{2 x^{2}+x -4}\right )}{4}-\frac {\ln \left (\frac {-131072-524288 x +65536 \sqrt {x^{4}-3 x^{2}+4}+30198 x^{20}-1296128 x^{9}-4373760 x^{10}+2095616 x^{7}-2336768 x^{5}-3865344 x^{6}-237568 x^{2}+1605632 x^{3}+4931072 x^{8}+1795584 x^{4}+624256 x^{11}+2875968 x^{12}+546720 x^{16}-128 x^{25}-3507 x^{22}+32744 x^{19}+16 x^{26}+1568 x^{23}+116 x^{24}-9128 x^{21}-154096 x^{18}-81008 x^{17}+156064 x^{15}-1437984 x^{14}-292096 x^{13}+16 \sqrt {x^{4}-3 x^{2}+4}\, x^{24}-128 \sqrt {x^{4}-3 x^{2}+4}\, x^{23}+140 \sqrt {x^{4}-3 x^{2}+4}\, x^{22}+1376 \sqrt {x^{4}-3 x^{2}+4}\, x^{21}-3311 \sqrt {x^{4}-3 x^{2}+4}\, x^{20}-6952 \sqrt {x^{4}-3 x^{2}+4}\, x^{19}+25088 \sqrt {x^{4}-3 x^{2}+4}\, x^{18}+21280 \sqrt {x^{4}-3 x^{2}+4}\, x^{17}-113776 \sqrt {x^{4}-3 x^{2}+4}\, x^{16}-44608 \sqrt {x^{4}-3 x^{2}+4}\, x^{15}+358208 \sqrt {x^{4}-3 x^{2}+4}\, x^{14}+77632 \sqrt {x^{4}-3 x^{2}+4}\, x^{13}-828960 \sqrt {x^{4}-3 x^{2}+4}\, x^{12}-155264 \sqrt {x^{4}-3 x^{2}+4}\, x^{11}+1432832 \sqrt {x^{4}-3 x^{2}+4}\, x^{10}+356864 \sqrt {x^{4}-3 x^{2}+4}\, x^{9}-1820416 \sqrt {x^{4}-3 x^{2}+4}\, x^{8}-680960 \sqrt {x^{4}-3 x^{2}+4}\, x^{7}+1605632 \sqrt {x^{4}-3 x^{2}+4}\, x^{6}+889856 \sqrt {x^{4}-3 x^{2}+4}\, x^{5}-847616 \sqrt {x^{4}-3 x^{2}+4}\, x^{4}-704512 \sqrt {x^{4}-3 x^{2}+4}\, x^{3}+143360 \sqrt {x^{4}-3 x^{2}+4}\, x^{2}+262144 \sqrt {x^{4}-3 x^{2}+4}\, x}{\left (x^{2}-2\right )^{8} x^{5}}\right )}{4}\)	\(646\)
risch	\(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}-\frac {5 \arcsinh \left (\frac {2 \sqrt {7}\, \left (x^{2}-\frac {3}{2}\right )}{7}\right )}{8}+\frac {9 \sqrt {1-\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2-6 i \sqrt {7}}}{4}\right )}{2 \sqrt {6+2 i \sqrt {7}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {25 \sqrt {33}\, \arctanh \left (-\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {13}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}}-\frac {25 \sqrt {33}\, \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}-\frac {25 \arctanh \left (-\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {13}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}}+\frac {25 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}+\frac {25 \sqrt {33}\, \arctanh \left (\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {13}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}}+\frac {25 \sqrt {33}\, \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}+\frac {25 \arctanh \left (\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {13}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}}+\frac {25 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}+\frac {5 \arctanh \left (\frac {-3 x^{2}+8}{4 \sqrt {x^{4}-3 x^{2}+4}}\right )}{8}+\frac {4 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}-3 x^{2}+4}}\)	\(1218\)
default	\(-\frac {3 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticF \left (\frac {x \sqrt {6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2-6 i \sqrt {7}}}{4}\right )}{2 \sqrt {6+2 i \sqrt {7}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {35 \ln \left (2 x^{2}-3+2 \sqrt {x^{4}-3 x^{2}+4}\right )}{32}-\frac {32 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticF \left (\frac {x \sqrt {6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2-6 i \sqrt {7}}}{4}\right )}{\sqrt {6+2 i \sqrt {7}}\, \sqrt {x^{4}-3 x^{2}+4}\, \left (i \sqrt {7}-3\right )}+\frac {32 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticE \left (\frac {x \sqrt {6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2-6 i \sqrt {7}}}{4}\right )}{\sqrt {6+2 i \sqrt {7}}\, \sqrt {x^{4}-3 x^{2}+4}\, \left (i \sqrt {7}-3\right )}+\frac {25 \sqrt {33}\, \arctanh \left (-\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {13}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}}-\frac {25 \sqrt {33}\, \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}-\frac {25 \arctanh \left (-\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {13}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}-\frac {5 \sqrt {33}}{32}}}+\frac {25 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}+\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}+\frac {25 \sqrt {33}\, \arctanh \left (\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {13}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}}+\frac {25 \sqrt {33}\, \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}+\frac {25 \arctanh \left (\frac {5 x^{2}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {x^{2} \sqrt {33}}{8 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {13}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}-\frac {3 \sqrt {33}}{16 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}\, \sqrt {x^{4}-3 x^{2}+4}}\right )}{64 \sqrt {\frac {85}{32}+\frac {5 \sqrt {33}}{32}}}+\frac {25 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right )^{2}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{32 \sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \left (-\frac {1}{4}-\frac {\sqrt {33}}{4}\right ) \sqrt {x^{4}-3 x^{2}+4}}+\frac {4 \sqrt {1-\frac {3 x^{2}}{8}-\frac {i x^{2} \sqrt {7}}{8}}\, \sqrt {1-\frac {3 x^{2}}{8}+\frac {i x^{2} \sqrt {7}}{8}}\, \EllipticPi \left (\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , \frac {1}{\frac {3}{4}+\frac {i \sqrt {7}}{4}}, \frac {\sqrt {\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{\sqrt {\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+\frac {6 \sqrt {1-\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2-6 i \sqrt {7}}}{4}\right )}{\sqrt {6+2 i \sqrt {7}}\, \sqrt {x^{4}-3 x^{2}+4}}+\frac {32 \sqrt {1-\left (\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2-6 i \sqrt {7}}}{4}\right )-\EllipticE \left (\frac {x \sqrt {6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2-6 i \sqrt {7}}}{4}\right )\right )}{\sqrt {6+2 i \sqrt {7}}\, \sqrt {x^{4}-3 x^{2}+4}\, \left (i \sqrt {7}-3\right )}+\frac {15 \arcsinh \left (\frac {2 \sqrt {7}\, \left (x^{2}-\frac {3}{2}\right )}{7}\right )}{32}+\frac {5 \arctanh \left (\frac {-3 x^{2}+8}{4 \sqrt {x^{4}-3 x^{2}+4}}\right )}{8}\)	\(1643\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*arcsinh(2/7*7^(1/2)*(x^2-3/2))-160/(17+33^(1/2))/(-17+33^(1/2))*arctanh(1/4*(-3*x^2+8)/(x^4-3*x^2+4)^(1/2
))-25/1056*(-297+25*33^(1/2))*33^(1/2)/(-17+33^(1/2))/(1/8*165^(1/2)-1/8*5^(1/2))*arctanh(4*(85/16-5/16*33^(1/
2)+(5/4-1/4*33^(1/2))*(x^2-17/8+1/8*33^(1/2)))/(1/8*165^(1/2)-1/8*5^(1/2))/(64*(x^2-17/8+1/8*33^(1/2))^2+64*(5
/4-1/4*33^(1/2))*(x^2-17/8+1/8*33^(1/2))+170-10*33^(1/2))^(1/2))+25/1056*(297+25*33^(1/2))*33^(1/2)/(17+33^(1/
2))/(1/8*165^(1/2)+1/8*5^(1/2))*arctanh(4*(85/16+5/16*33^(1/2)+(5/4+1/4*33^(1/2))*(x^2-17/8-1/8*33^(1/2)))/(1/
8*165^(1/2)+1/8*5^(1/2))/(64*(x^2-17/8-1/8*33^(1/2))^2+64*(5/4+1/4*33^(1/2))*(x^2-17/8-1/8*33^(1/2))+170+10*33
^(1/2))^(1/2))+1/2*(1/2*(x^4-3*x^2+4)^(1/2)*2^(1/2)/x-5/4*10^(1/2)*arctanh(1/5*10^(1/2)*(x^4-3*x^2+4)^(1/2)*2^
(1/2)/x)+2*2^(1/2)*arctanh((x^4-3*x^2+4)^(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 {\sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^2+2\right )\,\left (-x^2+2\,x+2\right )\,\sqrt {x^4-3\,x^2+4}}{x^2\,\left (x^2-2\right )\,\left (2\,x^2+x-4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 2\right ) \left (x^{2} - 2 x - 2\right ) \sqrt {x^{4} - 3 x^{2} + 4}}{x^{2} \left (x^{2} - 2\right ) \left (2 x^{2} + x - 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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