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

Optimal. Leaf size=37 \[ \frac{1}{4} \log \left (1-2 x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2 + Log[1 - 2*x^2]/4

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Rubi [A]  time = 0.0418386, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {6742, 260, 402, 216, 377, 207} \[ \frac{1}{4} \log \left (1-2 x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x - Sqrt[1 - x^2])^(-1),x]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2 + Log[1 - 2*x^2]/4

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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

Mathematica [A]  time = 0.0223991, size = 37, normalized size = 1. \[ \frac{1}{4} \log \left (1-2 x^2\right )-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{1-x^2}}\right )-\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(x - Sqrt[1 - x^2])^(-1),x]

[Out]

-ArcSin[x]/2 - ArcTanh[x/Sqrt[1 - x^2]]/2 + Log[1 - 2*x^2]/4

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Maple [B]  time = 0.031, size = 175, normalized size = 4.7 \begin{align*}{\frac{\ln \left ( 2\,{x}^{2}-1 \right ) }{4}}-{\frac{\sqrt{2}}{8}\sqrt{-4\, \left ( x+1/2\,\sqrt{2} \right ) ^{2}+4\, \left ( x+1/2\,\sqrt{2} \right ) \sqrt{2}+2}}-{\frac{\arcsin \left ( x \right ) }{2}}+{\frac{1}{4}{\it Artanh} \left ({\sqrt{2} \left ( 1+ \left ( x+{\frac{\sqrt{2}}{2}} \right ) \sqrt{2} \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 ) }+{\frac{\sqrt{2}}{8}\sqrt{-4\, \left ( x-1/2\,\sqrt{2} \right ) ^{2}-4\, \left ( x-1/2\,\sqrt{2} \right ) \sqrt{2}+2}}-{\frac{1}{4}{\it Artanh} \left ({\sqrt{2} \left ( 1- \left ( x-{\frac{\sqrt{2}}{2}} \right ) \sqrt{2} \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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*ln(2*x^2-1)-1/8*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*arcsin(x)+1/4*arctanh
((1+(x+1/2*2^(1/2))*2^(1/2))*2^(1/2)/(-4*(x+1/2*2^(1/2))^2+4*(x+1/2*2^(1/2))*2^(1/2)+2)^(1/2))+1/8*2^(1/2)*(-4
*(x-1/2*2^(1/2))^2-4*(x-1/2*2^(1/2))*2^(1/2)+2)^(1/2)-1/4*arctanh((1-(x-1/2*2^(1/2))*2^(1/2))*2^(1/2)/(-4*(x-1
/2*2^(1/2))^2-4*(x-1/2*2^(1/2))*2^(1/2)+2)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x - sqrt(-x^2 + 1)), x)

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Fricas [B]  time = 1.48303, size = 220, normalized size = 5.95 \begin{align*} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + \frac{1}{4} \, \log \left (2 \, x^{2} - 1\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(1/(x-(-x^2+1)^(1/2)),x, algorithm="fricas")

[Out]

arctan((sqrt(-x^2 + 1) - 1)/x) + 1/4*log(2*x^2 - 1) + 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 [A]  time = 0.152398, size = 17, normalized size = 0.46 \begin{align*} \frac{\log{\left (x - \sqrt{1 - x^{2}} \right )}}{2} - \frac{\operatorname{asin}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - sqrt(1 - x**2))/2 - asin(x)/2

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Giac [B]  time = 1.13184, size = 189, normalized size = 5.11 \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 | x + \frac{1}{2} \, \sqrt{2} \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x - \frac{1}{2} \, \sqrt{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 ) + \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(1/(x-(-x^2+1)^(1/2)),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 + 1/
2*sqrt(2))) + 1/4*log(abs(x - 1/2*sqrt(2))) - 1/4*log(abs(-x/(sqrt(-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))