3.73 \(\int \frac{\sqrt{1-x^2}}{-1+2 x^2} \, dx\)

Optimal. Leaf size=25 \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2

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Rubi [A]  time = 0.0128449, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {402, 216, 377, 207} \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(-1 + 2*x^2),x]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 216

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x^2}}{-1+2 x^2} \, dx &=-\left (\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\right )+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2} \left (-1+2 x^2\right )} \, dx\\ &=-\frac{1}{2} \sin ^{-1}(x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )\\ &=-\frac{1}{2} \sin ^{-1}(x)-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0101558, size = 25, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2

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Maple [B]  time = 0.029, size = 187, normalized size = 7.5 \begin{align*} -{\frac{\sqrt{2}}{2} \left ({\frac{1}{4}\sqrt{-4\, \left ( x+1/2\,\sqrt{2} \right ) ^{2}+4\, \left ( x+1/2\,\sqrt{2} \right ) \sqrt{2}+2}}+{\frac{\sqrt{2}\arcsin \left ( x \right ) }{4}}-{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\sqrt{2} \left ( \left ( x+{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}+1 \right ){\frac{1}{\sqrt{-4\, \left ( x+1/2\,\sqrt{2} \right ) ^{2}+4\, \left ( x+1/2\,\sqrt{2} \right ) \sqrt{2}+2}}}} \right ) } \right ) }+{\frac{\sqrt{2}}{2} \left ({\frac{1}{4}\sqrt{-4\, \left ( x-1/2\,\sqrt{2} \right ) ^{2}-4\,\sqrt{2} \left ( x-1/2\,\sqrt{2} \right ) +2}}-{\frac{\sqrt{2}\arcsin \left ( x \right ) }{4}}-{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\sqrt{2} \left ( -\sqrt{2} \left ( x-{\frac{\sqrt{2}}{2}} \right ) +1 \right ){\frac{1}{\sqrt{-4\, \left ( x-1/2\,\sqrt{2} \right ) ^{2}-4\,\sqrt{2} \left ( x-1/2\,\sqrt{2} \right ) +2}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*2^(1/2)*(1/4*(-4*(x+1/2*2^(1/2))^2+4*(x+1/2*2^(1/2))*2^(1/2)+2)^(1/2)+1/4*2^(1/2)*arcsin(x)-1/4*2^(1/2)*a
rctanh(((x+1/2*2^(1/2))*2^(1/2)+1)*2^(1/2)/(-4*(x+1/2*2^(1/2))^2+4*(x+1/2*2^(1/2))*2^(1/2)+2)^(1/2)))+1/2*2^(1
/2)*(1/4*(-4*(x-1/2*2^(1/2))^2-4*2^(1/2)*(x-1/2*2^(1/2))+2)^(1/2)-1/4*2^(1/2)*arcsin(x)-1/4*2^(1/2)*arctanh((-
2^(1/2)*(x-1/2*2^(1/2))+1)*2^(1/2)/(-4*(x-1/2*2^(1/2))^2-4*2^(1/2)*(x-1/2*2^(1/2))+2)^(1/2)))

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Maxima [B]  time = 1.51138, size = 149, normalized size = 5.96 \begin{align*} -\frac{1}{8} \, \sqrt{2}{\left (2 \, \sqrt{2} \arcsin \left (x\right ) - \sqrt{2} \log \left (\frac{1}{4} \, \sqrt{2} + \frac{\sqrt{2} \sqrt{-x^{2} + 1}}{{\left | 4 \, x + 2 \, \sqrt{2} \right |}} + \frac{1}{{\left | 4 \, x + 2 \, \sqrt{2} \right |}}\right ) + \sqrt{2} \log \left (-\frac{1}{4} \, \sqrt{2} + \frac{\sqrt{2} \sqrt{-x^{2} + 1}}{{\left | 4 \, x - 2 \, \sqrt{2} \right |}} + \frac{1}{{\left | 4 \, x - 2 \, \sqrt{2} \right |}}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*(2*sqrt(2)*arcsin(x) - sqrt(2)*log(1/4*sqrt(2) + sqrt(2)*sqrt(-x^2 + 1)/abs(4*x + 2*sqrt(2)) + 1/
abs(4*x + 2*sqrt(2))) + sqrt(2)*log(-1/4*sqrt(2) + sqrt(2)*sqrt(-x^2 + 1)/abs(4*x - 2*sqrt(2)) + 1/abs(4*x - 2
*sqrt(2))))

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Fricas [B]  time = 1.56759, size = 192, normalized size = 7.68 \begin{align*} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + \frac{1}{4} \, \log \left (-\frac{x^{2} + \sqrt{-x^{2} + 1}{\left (x + 1\right )} - x - 1}{x^{2}}\right ) - \frac{1}{4} \, \log \left (-\frac{x^{2} - \sqrt{-x^{2} + 1}{\left (x - 1\right )} + x - 1}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{2 x^{2} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.14134, size = 159, normalized size = 6.36 \begin{align*} -\frac{1}{4} \, \pi \mathrm{sgn}\left (x\right ) - \frac{1}{2} \, \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) - \frac{1}{4} \, \log \left ({\left | -\frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} + 2 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | -\frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*pi*sgn(x) - 1/2*arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1)) - 1/4*log(abs(-x/(sq
rt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x + 2)) + 1/4*log(abs(-x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/
x - 2))