### 3.824 $$\int \frac{\sqrt{1-x^2}}{(1-x)^2} \, dx$$

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

[Out]

(2*Sqrt[1 - x^2])/(1 - x) - ArcSin[x]

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Rubi [A]  time = 0.0068515, antiderivative size = 25, normalized size of antiderivative = 1., 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} $\frac{2 \sqrt{1-x^2}}{1-x}-\sin ^{-1}(x)$

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 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]

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]

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*}

Mathematica [A]  time = 0.0548329, size = 50, normalized size = 2. $2 \sqrt{1-x^2} \left (\frac{1}{1-x}+\frac{\sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )}{\sqrt{x-1} \sqrt{x+1}}\right )$

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]/(Sqrt[-1 + x]*Sqrt[1 + x]))

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Maple [A]  time = 0.045, size = 40, normalized size = 1.6 \begin{align*}{\frac{1}{ \left ( -1+x \right ) ^{2}} \left ( - \left ( -1+x \right ) ^{2}+2-2\,x \right ) ^{{\frac{3}{2}}}}+\sqrt{- \left ( -1+x \right ) ^{2}+2-2\,x}-\arcsin \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53006, size = 28, normalized size = 1.12 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \end{align*}

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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Fricas [A]  time = 2.14285, size = 104, normalized size = 4.16 \begin{align*} \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{align*}

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

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

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