\(\int \frac {1}{x^2 \sqrt {1-x^2}} \, dx\) [124]

Optimal result

Integrand size = 15, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]

[In]

Int[1/(x^2*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[1 - x^2]/x)

Rule 270

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*} \text {integral}& = -\frac {\sqrt {1-x^2}}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]

[In]

Integrate[1/(x^2*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[1 - x^2]/x)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {\sqrt {-x^{2}+1}}{x}\)	\(15\)
trager	\(-\frac {\sqrt {-x^{2}+1}}{x}\)	\(15\)
meijerg	\(-\frac {\sqrt {-x^{2}+1}}{x}\)	\(15\)
pseudoelliptic	\(-\frac {\sqrt {-x^{2}+1}}{x}\)	\(15\)
risch	\(\frac {x^{2}-1}{x \sqrt {-x^{2}+1}}\)	\(19\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right )}{x \sqrt {-x^{2}+1}}\)	\(20\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{x} \]

[In]

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

[Out]

-sqrt(-x^2 + 1)/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=\begin {cases} - \frac {i \sqrt {x^{2} - 1}}{x} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {\sqrt {1 - x^{2}}}{x} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{x} \]

[In]

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

[Out]

-sqrt(-x^2 + 1)/x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=\frac {x}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} - \frac {\sqrt {-x^{2} + 1} - 1}{2 \, x} \]

[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

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]

[In]

int(1/(x^2*(1 - x^2)^(1/2)),x)

[Out]

-(1 - x^2)^(1/2)/x