\(\int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx\) [48]

Optimal result

Integrand size = 22, antiderivative size = 46 \[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {5}+4 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

-1/3*arctan(1/3*(-4*x+5^(1/2))*3^(1/2))*3^(1/2)+1/3*arctan(1/3*(4*x+5^(1/2))*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1175, 632, 210} \[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=\frac {\arctan \left (\frac {4 x+\sqrt {5}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Int[(1 + 2*x^2)/(1 - x^2 + 4*x^4),x]

[Out]

-(ArcTan[(Sqrt[5] - 4*x)/Sqrt[3]]/Sqrt[3]) + ArcTan[(Sqrt[5] + 4*x)/Sqrt[3]]/Sqrt[3]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1175

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5} x}{2}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5} x}{2}+x^2} \, dx \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,-\frac {\sqrt {5}}{2}+2 x\right )\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,\frac {\sqrt {5}}{2}+2 x\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {5}+4 x}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.20 \[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=\frac {\left (-5 i+\sqrt {15}\right ) \arctan \left (\frac {2 x}{\sqrt {\frac {1}{2} \left (-1-i \sqrt {15}\right )}}\right )}{\sqrt {30 \left (-1-i \sqrt {15}\right )}}+\frac {\left (5 i+\sqrt {15}\right ) \arctan \left (\frac {2 x}{\sqrt {\frac {1}{2} \left (-1+i \sqrt {15}\right )}}\right )}{\sqrt {30 \left (-1+i \sqrt {15}\right )}} \]

[In]

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

[Out]

((-5*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(-1 - I*Sqrt[15])/2]])/Sqrt[30*(-1 - I*Sqrt[15])] + ((5*I + Sqrt[15])*Arc
Tan[(2*x)/Sqrt[(-1 + I*Sqrt[15])/2]])/Sqrt[30*(-1 + I*Sqrt[15])]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {\sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, \arctan \left (\frac {4 x^{3} \sqrt {3}}{3}+\frac {x \sqrt {3}}{3}\right )}{3}\)	\(35\)
default	\(\frac {\arctan \left (\frac {\left (4 x +\sqrt {5}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x -\sqrt {5}\right ) \sqrt {3}}{3}\right )}{3}\)	\(40\)

[In]

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

[Out]

1/3*3^(1/2)*arctan(2/3*x*3^(1/2))+1/3*3^(1/2)*arctan(4/3*x^3*3^(1/2)+1/3*x*3^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (4 \, x^{3} + x\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(4*x^3 + x)) + 1/3*sqrt(3)*arctan(2/3*sqrt(3)*x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=\frac {\sqrt {3} \cdot \left (2 \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} \right )} + 2 \operatorname {atan}{\left (\frac {4 \sqrt {3} x^{3}}{3} + \frac {\sqrt {3} x}{3} \right )}\right )}{6} \]

[In]

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

[Out]

sqrt(3)*(2*atan(2*sqrt(3)*x/3) + 2*atan(4*sqrt(3)*x**3/3 + sqrt(3)*x/3))/6

Maxima [F]

\[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=\int { \frac {2 \, x^{2} + 1}{4 \, x^{4} - x^{2} + 1} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {6} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x + \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {6} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x - \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(2/3*sqrt(6)*(1/4)^(3/4)*(4*x + sqrt(10)*(1/4)^(1/4))) + 1/3*sqrt(3)*arctan(2/3*sqrt(6)*(1/4
)^(3/4)*(4*x - sqrt(10)*(1/4)^(1/4)))

Mupad [B] (verification not implemented)

Time = 13.65 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.63 \[ \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx=\frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {4\,\sqrt {3}\,x^3}{3}+\frac {\sqrt {3}\,x}{3}\right )+\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}\right )\right )}{3} \]

[In]

int((2*x^2 + 1)/(4*x^4 - x^2 + 1),x)

[Out]

(3^(1/2)*(atan((3^(1/2)*x)/3 + (4*3^(1/2)*x^3)/3) + atan((2*3^(1/2)*x)/3)))/3