\(\int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx\) [825]

Optimal result

Integrand size = 19, antiderivative size = 22 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=\frac {\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {665} \[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=\frac {\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \]

[In]

Int[Sqrt[1 - x^2]/(1 - x)^3,x]

[Out]

(1 - x^2)^(3/2)/(3*(1 - x)^3)

Rule 665

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*} \text {integral}& = \frac {\left (1-x^2\right )^{3/2}}{3 (1-x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=-\frac {(-1-x) \sqrt {1-x^2}}{3 (-1+x)^2} \]

[In]

Integrate[Sqrt[1 - x^2]/(1 - x)^3,x]

[Out]

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

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
gosper	\(\frac {\left (1+x \right ) \sqrt {-x^{2}+1}}{3 \left (-1+x \right )^{2}}\)	\(20\)
trager	\(\frac {\left (1+x \right ) \sqrt {-x^{2}+1}}{3 \left (-1+x \right )^{2}}\)	\(20\)
default	\(-\frac {\left (-\left (-1+x \right )^{2}+2-2 x \right )^{\frac {3}{2}}}{3 \left (-1+x \right )^{3}}\)	\(22\)
risch	\(-\frac {x^{2}+2 x +1}{3 \left (-1+x \right ) \sqrt {-x^{2}+1}}\)	\(25\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (16) = 32\).

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=\frac {x^{2} + \sqrt {-x^{2} + 1} {\left (x + 1\right )} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

[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)

Sympy [F]

\[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=- \int \frac {\sqrt {1 - x^{2}}}{x^{3} - 3 x^{2} + 3 x - 1}\, dx \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).

Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=\frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (16) = 32\).

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \, {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} + 1\right )}^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx=\frac {\sqrt {1-x^2}\,\left (x+1\right )}{3\,{\left (x-1\right )}^2} \]

[In]

int(-(1 - x^2)^(1/2)/(x - 1)^3,x)

[Out]

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