\(\int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx\) [824]

Optimal result

Integrand size = 19, antiderivative size = 25 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=\frac {2 \sqrt {1-x^2}}{1-x}-\arcsin (x) \]

[Out]

-arcsin(x)+2*(-x^2+1)^(1/2)/(1-x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {677, 222} \[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=\frac {2 \sqrt {1-x^2}}{1-x}-\arcsin (x) \]

[In]

Int[Sqrt[1 - x^2]/(1 - x)^2,x]

[Out]

(2*Sqrt[1 - x^2])/(1 - x) - ArcSin[x]

Rule 222

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 677

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + p + 1))), x] - Dist[c*(p/(e^2*(m + p + 1))), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1
, 0] && IntegerQ[2*p]

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=-\frac {2 \sqrt {1-x^2}}{-1+x}+2 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]

[In]

Integrate[Sqrt[1 - x^2]/(1 - x)^2,x]

[Out]

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

Maple [A] (verified)

Time = 2.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {2+2 x}{\sqrt {-x^{2}+1}}-\arcsin \left (x \right )\)	\(20\)
default	\(\frac {\left (-\left (-1+x \right )^{2}+2-2 x \right )^{\frac {3}{2}}}{\left (-1+x \right )^{2}}+\sqrt {-\left (-1+x \right )^{2}+2-2 x}-\arcsin \left (x \right )\)	\(40\)
trager	\(-\frac {2 \sqrt {-x^{2}+1}}{-1+x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\)	\(45\)

[In]

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

[Out]

2*(1+x)/(-x^2+1)^(1/2)-arcsin(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=\frac {2 \, {\left ({\left (x - 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + x - \sqrt {-x^{2} + 1} - 1\right )}}{x - 1} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\left (x - 1\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \]

[In]

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

[Out]

-2*sqrt(-x^2 + 1)/(x - 1) - arcsin(x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=\text {Exception raised: NotImplementedError} \]

[In]

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

[Out]

Exception raised: NotImplementedError >> unable to parse Giac output: -(2*atan(i)-2*i)*sign((sageVARx-1)^-1)+2
*(-sqrt(-2*(sageVARx-1)^-1-1)*sign((sageVARx-1)^-1)+sign((sageVARx-1)^-1)*atan(sqrt(-2*(sageVARx-1)^-1-1)))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx=-\mathrm {asin}\left (x\right )-\frac {2\,\sqrt {1-x^2}}{x-1} \]

[In]

int((1 - x^2)^(1/2)/(x - 1)^2,x)

[Out]

- asin(x) - (2*(1 - x^2)^(1/2))/(x - 1)