\(\int \frac {1}{(1+x^2)^3 (1+x^2+x^4)^{3/2}} \, dx\) [243]

Optimal result

Integrand size = 20, antiderivative size = 190 \[ \int \frac {1}{\left (1+x^2\right )^3 \left (1+x^2+x^4\right )^{3/2}} \, dx=-\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}-\frac {x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac {3}{4} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {19 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{12 \sqrt {1+x^2+x^4}}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}} \]

[Out]

3/4*arctan(x/(x^4+x^2+1)^(1/2))-1/3*x*(-x^2+1)/(x^4+x^2+1)^(1/2)+1/4*x*(x^4+x^2+1)^(1/2)/(x^2+1)^2-1/3*x*(x^4+
x^2+1)^(1/2)/(x^2+1)+19/12*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*EllipticE(sin(2*arctan(x)),1/2)
*((x^4+x^2+1)/(x^2+1)^2)^(1/2)/(x^4+x^2+1)^(1/2)-5/4*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*Ellip
ticF(sin(2*arctan(x)),1/2)*((x^4+x^2+1)/(x^2+1)^2)^(1/2)/(x^4+x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {1242, 1106, 1211, 1117, 1209, 1237, 1710, 1607, 1726, 1714, 1712, 209, 12, 1331} \[ \int \frac {1}{\left (1+x^2\right )^3 \left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {3}{4} \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )-\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}+\frac {19 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{12 \sqrt {x^4+x^2+1}}-\frac {\sqrt {x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}-\frac {\left (1-x^2\right ) x}{3 \sqrt {x^4+x^2+1}} \]

[In]

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

[Out]

-1/3*(x*(1 - x^2))/Sqrt[1 + x^2 + x^4] + (x*Sqrt[1 + x^2 + x^4])/(4*(1 + x^2)^2) - (x*Sqrt[1 + x^2 + x^4])/(3*
(1 + x^2)) + (3*ArcTan[x/Sqrt[1 + x^2 + x^4]])/4 + (19*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2
*ArcTan[x], 1/4])/(12*Sqrt[1 + x^2 + x^4]) - (5*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan
[x], 1/4])/(4*Sqrt[1 + x^2 + x^4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

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

Rule 1106

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*
x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + 2*(p +
1)*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 -
4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1117

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)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1209

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)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], 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 1211

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 1237

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> Simp[(-e^2)*x*(d + e*x^2
)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b
*d*e + a*e^2)), Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*(q +
1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ
[b^2 - 4*a*c, 0] && ILtQ[q, -1]

Rule 1242

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{aa, bb, cc}, In
t[ExpandIntegrand[1/Sqrt[aa + bb*x^2 + cc*x^4], (d + e*x^2)^q*(aa + bb*x^2 + cc*x^4)^(p + 1/2), x] /. {aa -> a
, bb -> b, cc -> c}, 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]
&& ILtQ[q, 0] && IntegerQ[p + 1/2]

Rule 1331

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1710

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{A = Coeff
[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1
)*(Sqrt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(2*d*(q + 1)*(c*d^2 - b*d*e + a
*e^2)), Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d
- b*e)*(q + 1)) - 2*((B*d - A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 1)))*x^2 + c*(C*d^2 - B*d*
e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4]
&& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]

Rule 1712

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 1714

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

Rule 1726

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x,
x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-C/e^2, Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x],
x] + Dist[1/e^2, Int[(C*d^2 + A*e^2 + B*e^2*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a,
b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && 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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\left (1+x^2+x^4\right )^{3/2}}+\frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}}+\frac {1}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}}\right ) \, dx \\ & = -\int \frac {1}{\left (1+x^2+x^4\right )^{3/2}} \, dx+\int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx+\int \frac {1}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \, dx \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {x \sqrt {1+x^2+x^4}}{2 \left (1+x^2\right )}-\frac {1}{4} \int \frac {-3+2 x^2-x^4}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \, dx-\frac {1}{3} \int \frac {2+x^2}{\sqrt {1+x^2+x^4}} \, dx-\frac {1}{2} \int \frac {-1+2 x^2+x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {5 x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )}+\frac {1}{8} \int \frac {-10 x^2-6 x^4}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\frac {1}{3} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+\frac {1}{2} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx-\frac {1}{2} \int \frac {2 x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {5 x \sqrt {1+x^2+x^4}}{12 \left (1+x^2\right )}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{6 \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+\frac {1}{8} \int \frac {x^2 \left (-10-6 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\int \frac {x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {5 x \sqrt {1+x^2+x^4}}{12 \left (1+x^2\right )}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{6 \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+\frac {1}{8} \int \frac {-6-10 x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\frac {1}{2} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\frac {3}{4} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}-\frac {x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac {19 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{12 \sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}+\frac {1}{4} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )-\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}-\frac {x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {19 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{12 \sqrt {1+x^2+x^4}}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right ) \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}-\frac {x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac {3}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {19 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{12 \sqrt {1+x^2+x^4}}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.25 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (1+x^2\right )^3 \left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {4 x \left (-1+x^2\right ) \left (1+x^2\right )^2+3 x \left (1+x^2+x^4\right )+15 x \left (1+x^2\right ) \left (1+x^2+x^4\right )-\sqrt [3]{-1} \left (1+x^2\right )^2 \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (19 E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+\left (-9+10 i \sqrt {3}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )-18 \sqrt [3]{-1} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )}{12 \left (1+x^2\right )^2 \sqrt {1+x^2+x^4}} \]

[In]

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

[Out]

(4*x*(-1 + x^2)*(1 + x^2)^2 + 3*x*(1 + x^2 + x^4) + 15*x*(1 + x^2)*(1 + x^2 + x^4) - (-1)^(1/3)*(1 + x^2)^2*Sq
rt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(19*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + (-9 + (10
*I)*Sqrt[3])*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 18*(-1)^(1/3)*EllipticPi[(-1)^(1/3), I*ArcSinh[(
-1)^(5/6)*x], (-1)^(2/3)]))/(12*(1 + x^2)^2*Sqrt[1 + x^2 + x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.82

method	result	size
risch	\(\frac {x \left (19 x^{6}+37 x^{4}+29 x^{2}+14\right )}{12 \left (x^{2}+1\right )^{2} \sqrt {x^{4}+x^{2}+1}}-\frac {10 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {19 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(346\)
default	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )^{2}}+\frac {5 x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )}-\frac {2 \left (\frac {1}{6} x -\frac {1}{6} x^{3}\right )}{\sqrt {x^{4}+x^{2}+1}}-\frac {10 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {19 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {19 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(439\)
elliptic	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )^{2}}+\frac {5 x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )}-\frac {2 \left (\frac {1}{6} x -\frac {1}{6} x^{3}\right )}{\sqrt {x^{4}+x^{2}+1}}-\frac {10 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {19 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {19 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(439\)

[In]

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

[Out]

1/12*x*(19*x^6+37*x^4+29*x^2+14)/(x^2+1)^2/(x^4+x^2+1)^(1/2)-10/3/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2
))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*
(-2+2*I*3^(1/2))^(1/2))+19/3/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))
*x^2)^(1/2)/(x^4+x^2+1)^(1/2)/(1+I*3^(1/2))*(EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2)
)-EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2)))+3/2/(-1/2+1/2*I*3^(1/2))^(1/2)*(1+1/2*x^
2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^2+1/2*I*x^2*3^(1/2))^(1/2)/(x^4+x^2+1)^(1/2)*EllipticPi((-1/2+1/2*I*3^(1/2
))^(1/2)*x,-1/(-1/2+1/2*I*3^(1/2)),(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I*3^(1/2))^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\left (1+x^2\right )^3 \left (1+x^2+x^4\right )^{3/2}} \, dx=-\frac {19 \, \sqrt {2} {\left (x^{8} + 3 \, x^{6} + 4 \, x^{4} + 3 \, x^{2} - \sqrt {-3} {\left (x^{8} + 3 \, x^{6} + 4 \, x^{4} + 3 \, x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 2 \, \sqrt {2} {\left (15 \, x^{8} + 45 \, x^{6} + 60 \, x^{4} + 45 \, x^{2} - 4 \, \sqrt {-3} {\left (x^{8} + 3 \, x^{6} + 4 \, x^{4} + 3 \, x^{2} + 1\right )} + 15\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 36 \, {\left (x^{8} + 3 \, x^{6} + 4 \, x^{4} + 3 \, x^{2} + 1\right )} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) - 4 \, {\left (19 \, x^{7} + 37 \, x^{5} + 29 \, x^{3} + 14 \, x\right )} \sqrt {x^{4} + x^{2} + 1}}{48 \, {\left (x^{8} + 3 \, x^{6} + 4 \, x^{4} + 3 \, x^{2} + 1\right )}} \]

[In]

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

[Out]

-1/48*(19*sqrt(2)*(x^8 + 3*x^6 + 4*x^4 + 3*x^2 - sqrt(-3)*(x^8 + 3*x^6 + 4*x^4 + 3*x^2 + 1) + 1)*sqrt(sqrt(-3)
- 1)*elliptic_e(arcsin(1/2*sqrt(2)*x*sqrt(sqrt(-3) - 1)), 1/2*sqrt(-3) - 1/2) - 2*sqrt(2)*(15*x^8 + 45*x^6 +
60*x^4 + 45*x^2 - 4*sqrt(-3)*(x^8 + 3*x^6 + 4*x^4 + 3*x^2 + 1) + 15)*sqrt(sqrt(-3) - 1)*elliptic_f(arcsin(1/2*
sqrt(2)*x*sqrt(sqrt(-3) - 1)), 1/2*sqrt(-3) - 1/2) - 36*(x^8 + 3*x^6 + 4*x^4 + 3*x^2 + 1)*arctan(x/sqrt(x^4 +
x^2 + 1)) - 4*(19*x^7 + 37*x^5 + 29*x^3 + 14*x)*sqrt(x^4 + x^2 + 1))/(x^8 + 3*x^6 + 4*x^4 + 3*x^2 + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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