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

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.05 \begin {gather*} \frac {(x+1) \sqrt {1-x^2}}{3 (x-1)^2} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.27, size = 23, normalized size = 1.05 \begin {gather*} \frac {(x+1) \sqrt {1-x^2}}{3 (x-1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - x^2]/(1 - x)^3,x]

[Out]

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

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fricas [B]  time = 0.40, size = 33, normalized size = 1.50 \begin {gather*} \frac {x^{2} + \sqrt {-x^{2} + 1} {\left (x + 1\right )} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

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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giac [B]  time = 0.17, size = 41, normalized size = 1.86 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.04, size = 20, normalized size = 0.91 \begin {gather*} \frac {\left (x +1\right ) \sqrt {-x^{2}+1}}{3 \left (x -1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.32, size = 38, normalized size = 1.73 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.39, size = 19, normalized size = 0.86 \begin {gather*} \frac {\sqrt {1-x^2}\,\left (x+1\right )}{3\,{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1 - x^2)^(1/2)/(x - 1)^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {1 - x^{2}}}{x^{3} - 3 x^{2} + 3 x - 1}\, dx \end {gather*}

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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