\(\int \frac {(1+x^2)^3}{(1+x^2+x^4)^{3/2}} \, dx\) [238]

Optimal result

Integrand size = 20, antiderivative size = 144 \[ \int \frac {\left (1+x^2\right )^3}{\left (1+x^2+x^4\right )^{3/2}} \, dx=-\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {2 x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}-\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {1+x^2+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}} \]

[Out]

-1/3*x*(-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)+(x^2+1)*(cos(2*arc
tan(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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1219, 1211, 1117, 1209} \[ \int \frac {\left (1+x^2\right )^3}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\sqrt {x^4+x^2+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {x^4+x^2+1}}+\frac {2 \sqrt {x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac {\left (1-x^2\right ) x}{3 \sqrt {x^4+x^2+1}} \]

[In]

Int[(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] + (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]) + ((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]

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 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-x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {1}{3} \int \frac {4+2 x^2}{\sqrt {1+x^2+x^4}} \, dx \\ & = -\frac {x \left (1-x^2\right )}{3 \sqrt {1+x^2+x^4}}-\frac {2}{3} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+2 \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 {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}}+\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 )}{\sqrt {1+x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int \frac {\left (1+x^2\right )^3}{\left (1+x^2+x^4\right )^{3/2}} \, dx=\frac {-x+x^3+2 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+2 (-1)^{5/6} \sqrt {3+3 \sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{3 \sqrt {1+x^2+x^4}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.13 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.55

method	result	size
risch	\(\frac {x \left (x^{2}-1\right )}{3 \sqrt {x^{4}+x^{2}+1}}+\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(223\)
elliptic	\(-\frac {2 \left (\frac {1}{6} x -\frac {1}{6} x^{3}\right )}{\sqrt {x^{4}+x^{2}+1}}+\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(226\)
default	\(-\frac {4 \left (-\frac {1}{6} x +\frac {1}{6} x^{3}\right )}{\sqrt {x^{4}+x^{2}+1}}+\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-E\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {6 \left (\frac {1}{6} x^{3}+\frac {1}{3} x \right )}{\sqrt {x^{4}+x^{2}+1}}-\frac {6 \left (-\frac {1}{3} x^{3}-\frac {1}{6} x \right )}{\sqrt {x^{4}+x^{2}+1}}\)	\(268\)

[In]

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

[Out]

1/3*x*(x^2-1)/(x^4+x^2+1)^(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)*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 = 153, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1+x^2\right )^3}{\left (1+x^2+x^4\right )^{3/2}} \, dx=-\frac {\sqrt {2} {\left (x^{5} + x^{3} - \sqrt {-3} {\left (x^{5} + x^{3} + x\right )} + x\right )} \sqrt {\sqrt {-3} - 1} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-3} - 1}}{2 \, x}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - \sqrt {2} {\left (3 \, x^{5} + 3 \, x^{3} + \sqrt {-3} {\left (x^{5} + x^{3} + x\right )} + 3 \, x\right )} \sqrt {\sqrt {-3} - 1} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-3} - 1}}{2 \, x}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 2 \, {\left (3 \, x^{4} + x^{2} + 2\right )} \sqrt {x^{4} + x^{2} + 1}}{6 \, {\left (x^{5} + x^{3} + x\right )}} \]

[In]

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

[Out]

-1/6*(sqrt(2)*(x^5 + x^3 - sqrt(-3)*(x^5 + x^3 + x) + x)*sqrt(sqrt(-3) - 1)*elliptic_e(arcsin(1/2*sqrt(2)*sqrt
(sqrt(-3) - 1)/x), 1/2*sqrt(-3) - 1/2) - sqrt(2)*(3*x^5 + 3*x^3 + sqrt(-3)*(x^5 + x^3 + x) + 3*x)*sqrt(sqrt(-3
) - 1)*elliptic_f(arcsin(1/2*sqrt(2)*sqrt(sqrt(-3) - 1)/x), 1/2*sqrt(-3) - 1/2) - 2*(3*x^4 + x^2 + 2)*sqrt(x^4
+ x^2 + 1))/(x^5 + x^3 + x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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