### 3.823 $$\int \frac{\sqrt{1-x^2}}{1-x} \, dx$$

Optimal. Leaf size=16 $\sin ^{-1}(x)-\sqrt{1-x^2}$

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0064882, antiderivative size = 16, 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 = {665, 216} $\sin ^{-1}(x)-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[1 - x^2]/(1 - x),x]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

Rule 665

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 + 2*p + 1)), x] - Dist[(2*c*d*p)/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[
m + 2*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} \, dx &=-\sqrt{1-x^2}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x^2}+\sin ^{-1}(x)\\ \end{align*}

Mathematica [B]  time = 0.0495471, size = 44, normalized size = 2.75 $\frac{x^2+2 \sqrt{1-x^2} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-1}{\sqrt{1-x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[1 - x^2]/(1 - x),x]

[Out]

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

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Maple [A]  time = 0.04, size = 20, normalized size = 1.3 \begin{align*} -\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),x)

[Out]

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

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Maxima [A]  time = 1.70816, size = 19, normalized size = 1.19 \begin{align*} -\sqrt{-x^{2} + 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),x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 1) + arcsin(x)

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Fricas [B]  time = 2.21357, size = 70, normalized size = 4.38 \begin{align*} -\sqrt{-x^{2} + 1} - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.82402, size = 17, normalized size = 1.06 \begin{align*} - \begin{cases} \sqrt{1 - x^{2}} - \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((sqrt(1 - x**2) - asin(x), (x > -1) & (x < 1)))

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Giac [A]  time = 1.21201, size = 19, normalized size = 1.19 \begin{align*} -\sqrt{-x^{2} + 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),x, algorithm="giac")

[Out]

-sqrt(-x^2 + 1) + arcsin(x)