\(\int \frac {1-x^2}{(1+x^2) \sqrt {1+x^2+x^4}} \, dx\) [325]

Optimal result

Integrand size = 27, antiderivative size = 15 \[ \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1712, 209} \[ \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]

[In]

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

[Out]

ArcTan[x/Sqrt[1 + x^2 + x^4]]

Rule 209

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

Rule 1712

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[
A, Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B},
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] && EqQ[B*d + A*e, 0]

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]

[In]

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

[Out]

ArcTan[x/Sqrt[1 + x^2 + x^4]]

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
default	\(\arctan \left (\frac {x}{\sqrt {x^{4}+x^{2}+1}}\right )\)	\(14\)
pseudoelliptic	\(\arctan \left (\frac {x}{\sqrt {x^{4}+x^{2}+1}}\right )\)	\(14\)
elliptic	\(-\arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\)	\(18\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) \]

[In]

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

[Out]

arctan(x/sqrt(x^4 + x^2 + 1))

Sympy [F]

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

[In]

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

[Out]

-Integral(x**2/(x**2*sqrt(x**4 + x**2 + 1) + sqrt(x**4 + x**2 + 1)), x) - Integral(-1/(x**2*sqrt(x**4 + x**2 +
1) + sqrt(x**4 + x**2 + 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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