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

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {663, 216} \begin {gather*} \frac {2 \sqrt {1-x^2}}{1-x}-\sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(1 - x)^2,x]

[Out]

(2*Sqrt[1 - x^2])/(1 - x) - ArcSin[x]

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 663

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*} \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx &=\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*}

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Mathematica [A]  time = 0.06, size = 50, normalized size = 2.00 \begin {gather*} 2 \sqrt {1-x^2} \left (\frac {1}{1-x}+\frac {\sinh ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{\sqrt {x-1} \sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]/(1 - x)^2,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.24, size = 39, normalized size = 1.56 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right )-\frac {2 \sqrt {1-x^2}}{x-1} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - x^2]/(1 - x)^2,x]

[Out]

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

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fricas [A]  time = 0.40, size = 41, normalized size = 1.64 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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((x-1)^-1)+2*(-sqrt
(-2*(x-1)^-1-1)*sign((x-1)^-1)+sign((x-1)^-1)*atan(sqrt(-2*(x-1)^-1-1)))

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maple [A]  time = 0.04, size = 40, normalized size = 1.60 \begin {gather*} -\arcsin \relax (x )+\frac {\left (-2 x -\left (x -1\right )^{2}+2\right )^{\frac {3}{2}}}{\left (x -1\right )^{2}}+\sqrt {-2 x -\left (x -1\right )^{2}+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x-1)^2*(-2*x-(x-1)^2+2)^(3/2)+(-2*x-(x-1)^2+2)^(1/2)-arcsin(x)

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maxima [A]  time = 2.94, size = 21, normalized size = 0.84 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{x - 1} - \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-x^2 + 1)/(x - 1) - arcsin(x)

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mupad [B]  time = 0.04, size = 21, normalized size = 0.84 \begin {gather*} -\mathrm {asin}\relax (x)-\frac {2\,\sqrt {1-x^2}}{x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2)/(x - 1)^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\left (x - 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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