\(\int \frac {\sqrt {1+x^2+x^4}}{(1+x^2)^2} \, dx\) [229]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{2 \sqrt {1+x^2+x^4}} \]

[Out]

1/2*(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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1239} \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{2 \sqrt {x^4+x^2+1}} \]

[In]

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

[Out]

((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/(2*Sqrt[1 + x^2 + x^4])

Rule 1239

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

Rubi steps \begin{align*} \text {integral}& = \frac {\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 )}{2 \sqrt {1+x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 3.35 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^2} \, dx=\frac {\frac {x+x^3+x^5}{1+x^2}+(-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+\sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (-E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+\operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )}{2 \sqrt {1+x^2+x^4}} \]

[In]

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

[Out]

((x + x^3 + x^5)/(1 + x^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)] + (-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(-EllipticE[I*ArcSin
h[(-1)^(5/6)*x], (-1)^(2/3)] + EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/(2*Sqrt[1 + x^2 + x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 224, normalized size of antiderivative = 4.57

method	result	size
default	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{2 x^{2}+2}+\frac {\sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(224\)
risch	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{2 x^{2}+2}+\frac {\sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(224\)
elliptic	\(\frac {x \sqrt {x^{4}+x^{2}+1}}{2 x^{2}+2}+\frac {\sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(224\)

[In]

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

[Out]

1/2*x*(x^4+x^2+1)^(1/2)/(x^2+1)+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))+2/(-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)))

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^2\right )^2} \, dx=-\frac {2 \, \sqrt {2} \sqrt {-3} {\left (x^{2} + 1\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) + \sqrt {2} {\left (x^{2} - \sqrt {-3} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 4 \, \sqrt {x^{4} + x^{2} + 1} x}{8 \, {\left (x^{2} + 1\right )}} \]

[In]

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

[Out]

-1/8*(2*sqrt(2)*sqrt(-3)*(x^2 + 1)*sqrt(sqrt(-3) - 1)*elliptic_f(arcsin(1/2*sqrt(2)*x*sqrt(sqrt(-3) - 1)), 1/2
*sqrt(-3) - 1/2) + sqrt(2)*(x^2 - sqrt(-3)*(x^2 + 1) + 1)*sqrt(sqrt(-3) - 1)*elliptic_e(arcsin(1/2*sqrt(2)*x*s
qrt(sqrt(-3) - 1)), 1/2*sqrt(-3) - 1/2) - 4*sqrt(x^4 + x^2 + 1)*x)/(x^2 + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(x^4 + x^2 + 1)/(x^2 + 1)^2, x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(x^4 + x^2 + 1)/(x^2 + 1)^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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