\(\int \frac {1-2 x^2}{1+b x^2+4 x^4} \, dx\) [54]

Optimal result

Integrand size = 22, antiderivative size = 66 \[ \int \frac {1-2 x^2}{1+b x^2+4 x^4} \, dx=-\frac {\log \left (1-\sqrt {4-b} x+2 x^2\right )}{2 \sqrt {4-b}}+\frac {\log \left (1+\sqrt {4-b} x+2 x^2\right )}{2 \sqrt {4-b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1178, 642} \[ \int \frac {1-2 x^2}{1+b x^2+4 x^4} \, dx=\frac {\log \left (\sqrt {4-b} x+2 x^2+1\right )}{2 \sqrt {4-b}}-\frac {\log \left (-\sqrt {4-b} x+2 x^2+1\right )}{2 \sqrt {4-b}} \]

[In]

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

[Out]

-1/2*Log[1 - Sqrt[4 - b]*x + 2*x^2]/Sqrt[4 - b] + Log[1 + Sqrt[4 - b]*x + 2*x^2]/(2*Sqrt[4 - 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 {\frac {\sqrt {4-b}}{2}+2 x}{-\frac {1}{2}-\frac {1}{2} \sqrt {4-b} x-x^2} \, dx}{2 \sqrt {4-b}}-\frac {\int \frac {\frac {\sqrt {4-b}}{2}-2 x}{-\frac {1}{2}+\frac {1}{2} \sqrt {4-b} x-x^2} \, dx}{2 \sqrt {4-b}} \\ & = -\frac {\log \left (1-\sqrt {4-b} x+2 x^2\right )}{2 \sqrt {4-b}}+\frac {\log \left (1+\sqrt {4-b} x+2 x^2\right )}{2 \sqrt {4-b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.92 \[ \int \frac {1-2 x^2}{1+b x^2+4 x^4} \, dx=\frac {\frac {\left (4+b-\sqrt {-16+b^2}\right ) \arctan \left (\frac {2 \sqrt {2} x}{\sqrt {b-\sqrt {-16+b^2}}}\right )}{\sqrt {b-\sqrt {-16+b^2}}}-\frac {\left (4+b+\sqrt {-16+b^2}\right ) \arctan \left (\frac {2 \sqrt {2} x}{\sqrt {b+\sqrt {-16+b^2}}}\right )}{\sqrt {b+\sqrt {-16+b^2}}}}{\sqrt {2} \sqrt {-16+b^2}} \]

[In]

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

[Out]

(((4 + b - Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b - Sqrt[-16 + b^2]]])/Sqrt[b - Sqrt[-16 + b^2]] - ((4 +
b + Sqrt[-16 + b^2])*ArcTan[(2*Sqrt[2]*x)/Sqrt[b + Sqrt[-16 + b^2]]])/Sqrt[b + Sqrt[-16 + b^2]])/(Sqrt[2]*Sqr
t[-16 + b^2])

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18

method	result	size
risch	\(-\frac {\ln \left (-2 x^{2} \sqrt {4-b}+\left (4-b \right ) x -\sqrt {4-b}\right )}{2 \sqrt {4-b}}+\frac {\ln \left (-2 x^{2} \sqrt {4-b}+x \left (b -4\right )-\sqrt {4-b}\right )}{2 \sqrt {4-b}}\)	\(78\)
default	\(\frac {\left (-4-\sqrt {\left (b -4\right ) \left (4+b \right )}-b \right ) \arctan \left (\frac {4 x}{\sqrt {2 \sqrt {\left (b -4\right ) \left (4+b \right )}+2 b}}\right )}{\sqrt {\left (b -4\right ) \left (4+b \right )}\, \sqrt {2 \sqrt {\left (b -4\right ) \left (4+b \right )}+2 b}}+\frac {\left (4-\sqrt {\left (b -4\right ) \left (4+b \right )}+b \right ) \arctan \left (\frac {4 x}{\sqrt {-2 \sqrt {\left (b -4\right ) \left (4+b \right )}+2 b}}\right )}{\sqrt {\left (b -4\right ) \left (4+b \right )}\, \sqrt {-2 \sqrt {\left (b -4\right ) \left (4+b \right )}+2 b}}\)	\(128\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.65 \[ \int \frac {1-2 x^2}{1+b x^2+4 x^4} \, dx=\left [-\frac {\sqrt {-b + 4} \log \left (\frac {4 \, x^{4} - {\left (b - 8\right )} x^{2} + 2 \, {\left (2 \, x^{3} + x\right )} \sqrt {-b + 4} + 1}{4 \, x^{4} + b x^{2} + 1}\right )}{2 \, {\left (b - 4\right )}}, \frac {\sqrt {b - 4} \arctan \left (\frac {4 \, x^{3} + {\left (b - 2\right )} x}{\sqrt {b - 4}}\right ) - \sqrt {b - 4} \arctan \left (\frac {2 \, x}{\sqrt {b - 4}}\right )}{b - 4}\right ] \]

[In]

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

[Out]

[-1/2*sqrt(-b + 4)*log((4*x^4 - (b - 8)*x^2 + 2*(2*x^3 + x)*sqrt(-b + 4) + 1)/(4*x^4 + b*x^2 + 1))/(b - 4), (s
qrt(b - 4)*arctan((4*x^3 + (b - 2)*x)/sqrt(b - 4)) - sqrt(b - 4)*arctan(2*x/sqrt(b - 4)))/(b - 4)]

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.42 \[ \int \frac {1-2 x^2}{1+b x^2+4 x^4} \, dx=\frac {\sqrt {- \frac {1}{b - 4}} \log {\left (x^{2} + x \left (- \frac {b \sqrt {- \frac {1}{b - 4}}}{2} + 2 \sqrt {- \frac {1}{b - 4}}\right ) + \frac {1}{2} \right )}}{2} - \frac {\sqrt {- \frac {1}{b - 4}} \log {\left (x^{2} + x \left (\frac {b \sqrt {- \frac {1}{b - 4}}}{2} - 2 \sqrt {- \frac {1}{b - 4}}\right ) + \frac {1}{2} \right )}}{2} \]

[In]

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

[Out]

sqrt(-1/(b - 4))*log(x**2 + x*(-b*sqrt(-1/(b - 4))/2 + 2*sqrt(-1/(b - 4))) + 1/2)/2 - sqrt(-1/(b - 4))*log(x**
2 + x*(b*sqrt(-1/(b - 4))/2 - 2*sqrt(-1/(b - 4))) + 1/2)/2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {1-2 x^2}{1+b x^2+4 x^4} \, dx=-\frac {\mathrm {atan}\left (\frac {2\,x}{\sqrt {b-4}}\right )-\mathrm {atan}\left (\frac {b^3\,x+4\,b^2\,x^3-2\,b^2\,x-16\,b\,x-64\,x^3+32\,x}{{\left (b-4\right )}^{3/2}\,\left (b+4\right )}\right )}{\sqrt {b-4}} \]

[In]

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

[Out]

-(atan((2*x)/(b - 4)^(1/2)) - atan((32*x - 16*b*x - 2*b^2*x + b^3*x - 64*x^3 + 4*b^2*x^3)/((b - 4)^(3/2)*(b +
4))))/(b - 4)^(1/2)