\(\int \frac {1-x^2}{1+b x^2+x^4} \, dx\) [80]

Optimal result

Integrand size = 20, antiderivative size = 62 \[ \int \frac {1-x^2}{1+b x^2+x^4} \, dx=-\frac {\log \left (1-\sqrt {2-b} x+x^2\right )}{2 \sqrt {2-b}}+\frac {\log \left (1+\sqrt {2-b} x+x^2\right )}{2 \sqrt {2-b}} \]

[Out]

-1/2*ln(1+x^2-x*(2-b)^(1/2))/(2-b)^(1/2)+1/2*ln(1+x^2+x*(2-b)^(1/2))/(2-b)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1178, 642} \[ \int \frac {1-x^2}{1+b x^2+x^4} \, dx=\frac {\log \left (\sqrt {2-b} x+x^2+1\right )}{2 \sqrt {2-b}}-\frac {\log \left (-\sqrt {2-b} x+x^2+1\right )}{2 \sqrt {2-b}} \]

[In]

Int[(1 - x^2)/(1 + b*x^2 + x^4),x]

[Out]

-1/2*Log[1 - Sqrt[2 - b]*x + x^2]/Sqrt[2 - b] + Log[1 + Sqrt[2 - b]*x + x^2]/(2*Sqrt[2 - b])

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1178

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*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/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[b^2
- 4*a*c, 0]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(62)=124\).

Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.02 \[ \int \frac {1-x^2}{1+b x^2+x^4} \, dx=\frac {\frac {\left (2+b-\sqrt {-4+b^2}\right ) \arctan \left (\frac {\sqrt {2} x}{\sqrt {b-\sqrt {-4+b^2}}}\right )}{\sqrt {b-\sqrt {-4+b^2}}}-\frac {\left (2+b+\sqrt {-4+b^2}\right ) \arctan \left (\frac {\sqrt {2} x}{\sqrt {b+\sqrt {-4+b^2}}}\right )}{\sqrt {b+\sqrt {-4+b^2}}}}{\sqrt {2} \sqrt {-4+b^2}} \]

[In]

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

[Out]

(((2 + b - Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b - Sqrt[-4 + b^2]]])/Sqrt[b - Sqrt[-4 + b^2]] - ((2 + b +
Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b + Sqrt[-4 + b^2]]])/Sqrt[b + Sqrt[-4 + b^2]])/(Sqrt[2]*Sqrt[-4 + b^2
])

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.26

method	result	size
risch	\(-\frac {\ln \left (-x^{2} \sqrt {2-b}+\left (2-b \right ) x -\sqrt {2-b}\right )}{2 \sqrt {2-b}}+\frac {\ln \left (-x^{2} \sqrt {2-b}+x \left (b -2\right )-\sqrt {2-b}\right )}{2 \sqrt {2-b}}\)	\(78\)
default	\(\frac {\left (-2-\sqrt {\left (b -2\right ) \left (2+b \right )}-b \right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}\right )}{\sqrt {\left (b -2\right ) \left (2+b \right )}\, \sqrt {2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}+\frac {\left (2-\sqrt {\left (b -2\right ) \left (2+b \right )}+b \right ) \arctan \left (\frac {2 x}{\sqrt {-2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}\right )}{\sqrt {\left (b -2\right ) \left (2+b \right )}\, \sqrt {-2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}\)	\(128\)

[In]

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

[Out]

-1/2/(2-b)^(1/2)*ln(-x^2*(2-b)^(1/2)+(2-b)*x-(2-b)^(1/2))+1/2/(2-b)^(1/2)*ln(-x^2*(2-b)^(1/2)+x*(b-2)-(2-b)^(1
/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.61 \[ \int \frac {1-x^2}{1+b x^2+x^4} \, dx=\left [-\frac {\sqrt {-b + 2} \log \left (\frac {x^{4} - {\left (b - 4\right )} x^{2} + 2 \, {\left (x^{3} + x\right )} \sqrt {-b + 2} + 1}{x^{4} + b x^{2} + 1}\right )}{2 \, {\left (b - 2\right )}}, \frac {\sqrt {b - 2} \arctan \left (\frac {x^{3} + {\left (b - 1\right )} x}{\sqrt {b - 2}}\right ) - \sqrt {b - 2} \arctan \left (\frac {x}{\sqrt {b - 2}}\right )}{b - 2}\right ] \]

[In]

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

[Out]

[-1/2*sqrt(-b + 2)*log((x^4 - (b - 4)*x^2 + 2*(x^3 + x)*sqrt(-b + 2) + 1)/(x^4 + b*x^2 + 1))/(b - 2), (sqrt(b
- 2)*arctan((x^3 + (b - 1)*x)/sqrt(b - 2)) - sqrt(b - 2)*arctan(x/sqrt(b - 2)))/(b - 2)]

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.40 \[ \int \frac {1-x^2}{1+b x^2+x^4} \, dx=\frac {\sqrt {- \frac {1}{b - 2}} \log {\left (x^{2} + x \left (- b \sqrt {- \frac {1}{b - 2}} + 2 \sqrt {- \frac {1}{b - 2}}\right ) + 1 \right )}}{2} - \frac {\sqrt {- \frac {1}{b - 2}} \log {\left (x^{2} + x \left (b \sqrt {- \frac {1}{b - 2}} - 2 \sqrt {- \frac {1}{b - 2}}\right ) + 1 \right )}}{2} \]

[In]

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

[Out]

sqrt(-1/(b - 2))*log(x**2 + x*(-b*sqrt(-1/(b - 2)) + 2*sqrt(-1/(b - 2))) + 1)/2 - sqrt(-1/(b - 2))*log(x**2 +
x*(b*sqrt(-1/(b - 2)) - 2*sqrt(-1/(b - 2))) + 1)/2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int \frac {1-x^2}{1+b x^2+x^4} \, dx=-\frac {\mathrm {atan}\left (\frac {x}{\sqrt {b-2}}\right )-\mathrm {atan}\left (\left (b-2\right )\,\left (x\,\left (\frac {1}{\sqrt {b-2}}+\frac {\frac {4}{b-2}+1}{\sqrt {b-2}\,\left (b+2\right )}\right )+\frac {x^3\,\left (\frac {2\,b}{b-2}-1\right )}{\sqrt {b-2}\,\left (b+2\right )}\right )\right )}{\sqrt {b-2}} \]

[In]

int(-(x^2 - 1)/(b*x^2 + x^4 + 1),x)

[Out]

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