### 3.825 $$\int \frac{\sqrt{1-x^2}}{(1-x)^3} \, dx$$

Optimal. Leaf size=22 $\frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3}$

[Out]

(1 - x^2)^(3/2)/(3*(1 - x)^3)

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Rubi [A]  time = 0.0057352, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {651} $\frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[1 - x^2]/(1 - x)^3,x]

[Out]

(1 - x^2)^(3/2)/(3*(1 - x)^3)

Rule 651

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

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x^2}}{(1-x)^3} \, dx &=\frac{\left (1-x^2\right )^{3/2}}{3 (1-x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0167045, size = 23, normalized size = 1.05 $\frac{(x+1) \sqrt{1-x^2}}{3 (x-1)^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[1 - x^2]/(1 - x)^3,x]

[Out]

((1 + x)*Sqrt[1 - x^2])/(3*(-1 + x)^2)

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Maple [A]  time = 0.041, size = 20, normalized size = 0.9 \begin{align*}{\frac{1+x}{3\, \left ( -1+x \right ) ^{2}}\sqrt{-{x}^{2}+1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)^(1/2)/(1-x)^3,x)

[Out]

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

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Maxima [B]  time = 0.994955, size = 51, normalized size = 2.32 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1-x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.11562, size = 84, normalized size = 3.82 \begin{align*} \frac{x^{2} + \sqrt{-x^{2} + 1}{\left (x + 1\right )} - 2 \, x + 1}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1-x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sqrt{1 - x^{2}}}{x^{3} - 3 x^{2} + 3 x - 1}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+1)**(1/2)/(1-x)**3,x)

[Out]

-Integral(sqrt(1 - x**2)/(x**3 - 3*x**2 + 3*x - 1), x)

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Giac [B]  time = 1.2547, size = 55, normalized size = 2.5 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \,{\left (\frac{\sqrt{-x^{2} + 1} - 1}{x} + 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1-x)^3,x, algorithm="giac")

[Out]

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