### 3.124 $$\int \frac{1}{x^2 \sqrt{1-x^2}} \, dx$$

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

[Out]

-(Sqrt[1 - x^2]/x)

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Rubi [A]  time = 0.0031218, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {264} $-\frac{\sqrt{1-x^2}}{x}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^2*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[1 - x^2]/x)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{1-x^2}} \, dx &=-\frac{\sqrt{1-x^2}}{x}\\ \end{align*}

Mathematica [A]  time = 0.002725, size = 16, normalized size = 1. $-\frac{\sqrt{1-x^2}}{x}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x^2*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[1 - x^2]/x)

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Maple [A]  time = 0.003, size = 20, normalized size = 1.3 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) }{x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.39644, size = 19, normalized size = 1.19 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)/x

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Fricas [A]  time = 1.84782, size = 26, normalized size = 1.62 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)/x

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Sympy [A]  time = 0.708029, size = 27, normalized size = 1.69 \begin{align*} \begin{cases} - \frac{i \sqrt{x^{2} - 1}}{x} & \text{for}\: \left |{x^{2}}\right | > 1 \\- \frac{\sqrt{1 - x^{2}}}{x} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*sqrt(x**2 - 1)/x, Abs(x**2) > 1), (-sqrt(1 - x**2)/x, True))

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Giac [B]  time = 1.06196, size = 45, normalized size = 2.81 \begin{align*} \frac{x}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{\sqrt{-x^{2} + 1} - 1}{2 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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