3.3.64 \(\int \frac {x}{\sqrt {1-x^2}} \, dx\) [264]

Optimal. Leaf size=13 \[ -\sqrt {1-x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \begin {gather*} -\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2]

Rule 267

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2]

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Maple [A]
time = 0.04, size = 12, normalized size = 0.92

method	result	size
derivativedivides	\(-\sqrt {-x^{2}+1}\)	\(12\)
default	\(-\sqrt {-x^{2}+1}\)	\(12\)
trager	\(-\sqrt {-x^{2}+1}\)	\(12\)
risch	\(\frac {x^{2}-1}{\sqrt {-x^{2}+1}}\)	\(16\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right )}{\sqrt {-x^{2}+1}}\)	\(17\)
meijerg	\(-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.37, size = 11, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Fricas [A]
time = 0.45, size = 11, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Sympy [A]
time = 0.05, size = 8, normalized size = 0.62 \begin {gather*} - \sqrt {1 - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 - x**2)

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Giac [A]
time = 0.46, size = 11, normalized size = 0.85 \begin {gather*} -\sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Mupad [B]
time = 0.00, size = 11, normalized size = 0.85 \begin {gather*} -\sqrt {1-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1 - x^2)^(1/2),x)

[Out]

-(1 - x^2)^(1/2)

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