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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} -\frac {\sqrt {1-x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[1 - x^2]/x)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.13, size = 35, normalized size = 2.19 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {I \sqrt {-1+x^2}}{x},\text {Abs}\left [x^2\right ]>1\right \}\right \},-\frac {\sqrt {1-x^2}}{x}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^2*Sqrt[1 - x^2]),x]')

[Out]

Piecewise[{{-I Sqrt[-1 + x ^ 2] / x, Abs[x ^ 2] > 1}}, -Sqrt[1 - x ^ 2] / x]

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Maple [A]
time = 0.04, size = 15, normalized size = 0.94

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 14, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x} \end {gather*}

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 = 0.33, size = 14, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x} \end {gather*}

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 = 0.38, size = 27, normalized size = 1.69 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
time = 0.00, size = 40, normalized size = 2.50 \begin {gather*} -\frac {x}{-2 \sqrt {-x^{2}+1}+2}+\frac {-2 \sqrt {-x^{2}+1}+2}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.23, size = 14, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {1-x^2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(1 - x^2)^(1/2)),x)

[Out]

-(1 - x^2)^(1/2)/x

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