Integrand size = 20, antiderivative size = 142 \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\frac {x \sqrt {1+x^2+x^4}}{4 \left (1+x^2\right )^2}+\frac {1}{4} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{4 \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}} \]
1/4*arctan(x/(x^4+x^2+1)^(1/2))+1/4*x*(x^4+x^2+1)^(1/2)/(x^2+1)^2+3/4*(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)-1/2*(x^2+1)*(cos(2*a rctan(x))^2)^(1/2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),1/2)*((x^4+ x^2+1)/(x^2+1)^2)^(1/2)/(x^4+x^2+1)^(1/2)
Result contains complex when optimal does not.
Time = 10.26 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\frac {\frac {x \left (4+3 x^2\right ) \left (1+x^2+x^4\right )}{\left (1+x^2\right )^2}-3 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )-2 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+2 (-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{4 \sqrt {1+x^2+x^4}} \]
((x*(4 + 3*x^2)*(1 + x^2 + x^4))/(1 + x^2)^2 - 3*(-1)^(1/3)*Sqrt[1 + (-1)^ (1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (- 1)^(2/3)] - EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]) - 2*(-1)^(2/3) *Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1 )^(5/6)*x], (-1)^(2/3)] + 2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - ( -1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] )/(4*Sqrt[1 + x^2 + x^4])
Time = 0.68 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.30, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {1551, 25, 2210, 27, 2027, 2230, 1509, 2214, 1416, 2212, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (x^2+1\right )^3 \sqrt {x^4+x^2+1}} \, dx\) |
| \(\Big \downarrow \) 1551 |
| \(\displaystyle \frac {x \sqrt {x^4+x^2+1}}{4 \left (x^2+1\right )^2}-\frac {1}{4} \int -\frac {x^4-2 x^2+3}{\left (x^2+1\right )^2 \sqrt {x^4+x^2+1}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \int \frac {x^4-2 x^2+3}{\left (x^2+1\right )^2 \sqrt {x^4+x^2+1}}dx+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2210 |
| \(\displaystyle \frac {1}{4} \left (\frac {3 x \sqrt {x^4+x^2+1}}{x^2+1}-\frac {1}{2} \int \frac {2 \left (3 x^4+5 x^2\right )}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {3 x \sqrt {x^4+x^2+1}}{x^2+1}-\int \frac {3 x^4+5 x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \frac {1}{4} \left (\frac {3 x \sqrt {x^4+x^2+1}}{x^2+1}-\int \frac {x^2 \left (3 x^2+5\right )}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2230 |
| \(\displaystyle \frac {1}{4} \left (3 \int \frac {1-x^2}{\sqrt {x^4+x^2+1}}dx-\int \frac {5 x^2+3}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+\frac {3 \sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{4} \left (-\int \frac {5 x^2+3}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {3 \sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2214 |
| \(\displaystyle \frac {1}{4} \left (-4 \int \frac {1}{\sqrt {x^4+x^2+1}}dx+\int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx+3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {3 \sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\sqrt {x^4+x^2+1}}+3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {3 \sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2212 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {1}{\frac {x^2}{x^4+x^2+1}+1}d\frac {x}{\sqrt {x^4+x^2+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\sqrt {x^4+x^2+1}}+3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {3 \sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{4} \left (\arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\sqrt {x^4+x^2+1}}+3 \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {3 \sqrt {x^4+x^2+1} x}{x^2+1}\right )+\frac {\sqrt {x^4+x^2+1} x}{4 \left (x^2+1\right )^2}\) |
(x*Sqrt[1 + x^2 + x^4])/(4*(1 + x^2)^2) + ((3*x*Sqrt[1 + x^2 + x^4])/(1 + x^2) + ArcTan[x/Sqrt[1 + x^2 + x^4]] + 3*(-((x*Sqrt[1 + x^2 + x^4])/(1 + x ^2)) + ((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/Sqrt[1 + x^2 + x^4]) - (2*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^ 2]*EllipticF[2*ArcTan[x], 1/4])/Sqrt[1 + x^2 + x^4])/4
3.3.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> 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]
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] + Simp[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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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 = C oeff[P4x, x, 4]}, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1)*(Sq rt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[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] && ILtQ[q, -1 ]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Simp[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] & & EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Simp[(B*d + A*e)/(2*d*e) Int[1/Sqrt[a + b*x ^2 + c*x^4], x], x] - Simp[(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] && E qQ[c*d^2 - a*e^2, 0] && NeQ[B*d + A*e, 0]
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]}, Simp[-C/e^2 Int[(d - e*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[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] && Eq Q[c*d^2 - a*e^2, 0]
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.37
| method | result | size |
| risch | \(\frac {\sqrt {x^{4}+x^{2}+1}\, x \left (3 x^{2}+4\right )}{4 \left (x^{2}+1\right )^{2}}-\frac {\sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {3 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(336\) |
| default | \(\frac {x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )^{2}}+\frac {3 x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(418\) |
| elliptic | \(\frac {x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )^{2}}+\frac {3 x \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{2}+1\right )}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {3 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \Pi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(418\) |
1/4*(x^4+x^2+1)^(1/2)*x*(3*x^2+4)/(x^2+1)^2-1/(-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) )+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)))+1/2/(-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))
Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=-\frac {3 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} - \sqrt {-3} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 2 \, \sqrt {2} {\left (2 \, x^{4} + 4 \, x^{2} - \sqrt {-3} {\left (x^{4} + 2 \, x^{2} + 1\right )} + 2\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) - 4 \, \sqrt {x^{4} + x^{2} + 1} {\left (3 \, x^{3} + 4 \, x\right )}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
-1/16*(3*sqrt(2)*(x^4 + 2*x^2 - sqrt(-3)*(x^4 + 2*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)*(2*x^4 + 4*x^2 - sqrt(-3)*(x^4 + 2*x^2 + 1) + 2)*sqrt( sqrt(-3) - 1)*elliptic_f(arcsin(1/2*sqrt(2)*x*sqrt(sqrt(-3) - 1)), 1/2*sqr t(-3) - 1/2) - 4*(x^4 + 2*x^2 + 1)*arctan(x/sqrt(x^4 + x^2 + 1)) - 4*sqrt( x^4 + x^2 + 1)*(3*x^3 + 4*x))/(x^4 + 2*x^2 + 1)
\[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int \frac {1}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{3}}\, dx \]
\[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{3}} \,d x } \]
\[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int { \frac {1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (1+x^2\right )^3 \sqrt {1+x^2+x^4}} \, dx=\int \frac {1}{{\left (x^2+1\right )}^3\,\sqrt {x^4+x^2+1}} \,d x \]