\(\int \frac {1+x^2}{1-x^2+x^4} \, dx\) [75]

Optimal result

Integrand size = 18, antiderivative size = 23 \[ \int \frac {1+x^2}{1-x^2+x^4} \, dx=-\arctan \left (\sqrt {3}-2 x\right )+\arctan \left (\sqrt {3}+2 x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1175, 632, 210} \[ \int \frac {1+x^2}{1-x^2+x^4} \, dx=\arctan \left (2 x+\sqrt {3}\right )-\arctan \left (\sqrt {3}-2 x\right ) \]

[In]

Int[(1 + x^2)/(1 - x^2 + x^4),x]

[Out]

-ArcTan[Sqrt[3] - 2*x] + ArcTan[Sqrt[3] + 2*x]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1+x^2}{1-x^2+x^4} \, dx=-\arctan \left (\frac {x}{-1+x^2}\right ) \]

[In]

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

[Out]

-ArcTan[x/(-1 + x^2)]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.35

method	result	size
risch	\(\arctan \left (x^{3}\right )+\arctan \left (x \right )\)	\(8\)
default	\(\arctan \left (2 x -\sqrt {3}\right )+\arctan \left (2 x +\sqrt {3}\right )\)	\(20\)
parallelrisch	\(\frac {i \ln \left (x^{2}+i x -1\right )}{2}-\frac {i \ln \left (x^{2}-i x -1\right )}{2}\)	\(28\)

[In]

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

[Out]

arctan(x^3)+arctan(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.30 \[ \int \frac {1+x^2}{1-x^2+x^4} \, dx=\arctan \left (x^{3}\right ) + \arctan \left (x\right ) \]

[In]

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

[Out]

arctan(x^3) + arctan(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.30 \[ \int \frac {1+x^2}{1-x^2+x^4} \, dx=\operatorname {atan}{\left (x \right )} + \operatorname {atan}{\left (x^{3} \right )} \]

[In]

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

[Out]

atan(x) + atan(x**3)

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {1+x^2}{1-x^2+x^4} \, dx=\frac {1}{4} \, \pi \mathrm {sgn}\left (x\right ) + \frac {1}{2} \, \arctan \left (\frac {x^{4} - 3 \, x^{2} + 1}{2 \, {\left (x^{3} - x\right )}}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.28 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.30 \[ \int \frac {1+x^2}{1-x^2+x^4} \, dx=\mathrm {atan}\left (x^3\right )+\mathrm {atan}\left (x\right ) \]

[In]

int((x^2 + 1)/(x^4 - x^2 + 1),x)

[Out]

atan(x^3) + atan(x)