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

Optimal. Leaf size=62 \[ \frac{\log \left (\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}}-\frac{\log \left (-\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}} \]

[Out]

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

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Rubi [A]  time = 0.0289177, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1164, 628} \[ \frac{\log \left (\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}}-\frac{\log \left (-\sqrt{2-b} x+x^2+1\right )}{2 \sqrt{2-b}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 + b*x^2 + x^4),x]

[Out]

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

Rule 1164

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]

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{1-x^2}{1+b x^2+x^4} \, dx &=-\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]  time = 0.0711127, size = 125, normalized size = 2.02 \[ \frac{\frac{\left (-\sqrt{b^2-4}+b+2\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right )}{\sqrt{b-\sqrt{b^2-4}}}-\frac{\left (\sqrt{b^2-4}+b+2\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right )}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}} \]

Antiderivative was successfully verified.

[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
])

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Maple [B]  time = 0.105, size = 279, normalized size = 4.5 \begin{align*} -2\,{\frac{1}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ) }-{\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}}-{b\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}}+2\,{\frac{1}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ) }-{\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}}+{b\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{-2\,\sqrt{ \left ( -2+b \right ) \left ( 2+b \right ) }+2\,b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/((-2+b)*(2+b))^(1/2)/(2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2))-1/(2
*((-2+b)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2))-1/((-2+b)*(2+b))^(1/2)/(2*((-2
+b)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2))*b+2/((-2+b)*(2+b))^(1/2)/(-2*((-2+b
)*(2+b))^(1/2)+2*b)^(1/2)*arctan(2*x/(-2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2))-1/(-2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2
)*arctan(2*x/(-2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2))+1/((-2+b)*(2+b))^(1/2)/(-2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2)*a
rctan(2*x/(-2*((-2+b)*(2+b))^(1/2)+2*b)^(1/2))*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 1}{x^{4} + b x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.35622, size = 273, normalized size = 4.4 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [A]  time = 0.251893, size = 87, normalized size = 1.4 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError