Integrand size = 20, antiderivative size = 98 \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {x \left (1+2 x^2\right )}{3 \sqrt {1+x^2+x^4}}-\frac {2 x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {1+x^2+x^4}} \]
1/3*x*(2*x^2+1)/(x^4+x^2+1)^(1/2)-2/3*x*(x^4+x^2+1)^(1/2)/(x^2+1)+2/3*(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)
Result contains complex when optimal does not.
Time = 10.13 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.61 \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {x+2 x^3-2 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-i \sqrt {2+\left (1+i \sqrt {3}\right ) x^2} \sqrt {6+\left (3-3 i \sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \left (x+i \sqrt {3} x\right )\right ),\frac {1}{2} i \left (i+\sqrt {3}\right )\right )}{3 \sqrt {1+x^2+x^4}} \]
(x + 2*x^3 - 2*(-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)] - I*Sqrt[2 + (1 + I*Sqrt[ 3])*x^2]*Sqrt[6 + (3 - (3*I)*Sqrt[3])*x^2]*EllipticF[ArcSin[(x + I*Sqrt[3] *x)/2], (I/2)*(I + Sqrt[3])])/(3*Sqrt[1 + x^2 + x^4])
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1517, 27, 1509}
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.
| \(\displaystyle \int \frac {\left (x^2+1\right )^2}{\left (x^4+x^2+1\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {1}{3} \int \frac {2 \left (1-x^2\right )}{\sqrt {x^4+x^2+1}}dx+\frac {x \left (2 x^2+1\right )}{3 \sqrt {x^4+x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \int \frac {1-x^2}{\sqrt {x^4+x^2+1}}dx+\frac {x \left (2 x^2+1\right )}{3 \sqrt {x^4+x^2+1}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2}{3} \left (\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}-\frac {x \sqrt {x^4+x^2+1}}{x^2+1}\right )+\frac {x \left (2 x^2+1\right )}{3 \sqrt {x^4+x^2+1}}\) |
(x*(1 + 2*x^2))/(3*Sqrt[1 + x^2 + x^4]) + (2*(-((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]))/3
3.3.39.3.1 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[((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_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, 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] && IGtQ[q, 1] && LtQ[p, -1]
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.30
| method | result | size |
| risch | \(\frac {x \left (2 x^{2}+1\right )}{3 \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\) | \(225\) |
| elliptic | \(-\frac {2 \left (-\frac {1}{3} x^{3}-\frac {1}{6} x \right )}{\sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\) | \(226\) |
| default | \(-\frac {2 \left (-\frac {1}{6} x +\frac {1}{6} x^{3}\right )}{\sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \left (\frac {1}{6} x^{3}+\frac {1}{3} x \right )}{\sqrt {x^{4}+x^{2}+1}}-\frac {4 \left (-\frac {1}{3} x^{3}-\frac {1}{6} x \right )}{\sqrt {x^{4}+x^{2}+1}}\) | \(268\) |
1/3*x*(2*x^2+1)/(x^4+x^2+1)^(1/2)+4/3/(-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))+8/3/(- 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)))
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.34 \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {-3} {\left (x^{4} + x^{2} + 1\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) + \sqrt {2} {\left (x^{4} + x^{2} - \sqrt {-3} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{3} + x\right )}}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \]
-1/6*(2*sqrt(2)*sqrt(-3)*(x^4 + x^2 + 1)*sqrt(sqrt(-3) - 1)*elliptic_f(arc sin(1/2*sqrt(2)*x*sqrt(sqrt(-3) - 1)), 1/2*sqrt(-3) - 1/2) + sqrt(2)*(x^4 + x^2 - sqrt(-3)*(x^4 + 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(x^4 + x^2 + 1)*(2*x^3 + x))/(x^4 + x^2 + 1)
\[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\int \frac {\left (x^{2} + 1\right )^{2}}{\left (\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{2}}{{\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{2}}{{\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\int \frac {{\left (x^2+1\right )}^2}{{\left (x^4+x^2+1\right )}^{3/2}} \,d x \]