\(\int \frac {(1+x^2)^2}{(1+x^2+x^4)^{3/2}} \, dx\) [239]

Optimal result

Integrand size = 20, antiderivative size = 98 \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {x \left (1+2 x^2\right )}{3 \sqrt {1+x^2+x^4}}-\frac {2 x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {1+x^2+x^4}} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1219, 1209} \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {x^4+x^2+1}}-\frac {2 \sqrt {x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac {\left (2 x^2+1\right ) x}{3 \sqrt {x^4+x^2+1}} \]

[In]

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

[Out]

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

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 1219

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(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] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[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]

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (1+2 x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {1}{3} \int \frac {2-2 x^2}{\sqrt {1+x^2+x^4}} \, 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 \tan ^{-1}(x)|\frac {1}{4}\right )}{3 \sqrt {1+x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.13 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.61 \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {x+2 x^3-2 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-i \sqrt {2+\left (1+i \sqrt {3}\right ) x^2} \sqrt {6+\left (3-3 i \sqrt {3}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \left (x+i \sqrt {3} x\right )\right ),\frac {1}{2} i \left (i+\sqrt {3}\right )\right )}{3 \sqrt {1+x^2+x^4}} \]

[In]

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

[Out]

(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])

Maple [C] (verified)

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	\(-\frac {2 \left (-\frac {1}{6} x +\frac {1}{6} x^{3}\right )}{\sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \left (\frac {1}{6} x^{3}+\frac {1}{3} x \right )}{\sqrt {x^{4}+x^{2}+1}}-\frac {4 \left (-\frac {1}{3} x^{3}-\frac {1}{6} x \right )}{\sqrt {x^{4}+x^{2}+1}}\)	\(268\)

[In]

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

[Out]

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)))

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.34 \[ \int \frac {\left (1+x^2\right )^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {2} \sqrt {-3} {\left (x^{4} + x^{2} + 1\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) + \sqrt {2} {\left (x^{4} + x^{2} - \sqrt {-3} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {-3} - 1}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{3} + x\right )}}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \]

[In]

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

[Out]

-1/6*(2*sqrt(2)*sqrt(-3)*(x^4 + 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^4 + x^2 - sqrt(-3)*(x^4 + x^2 + 1) + 1)*sqrt(sqrt(-3) - 1)*elliptic_e(arcs
in(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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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