### 3.126 $$\int \frac{x}{\sqrt{1-x^2}} \, dx$$

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

[Out]

-Sqrt[1 - x^2]

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Rubi [A]  time = 0.0024715, antiderivative size = 13, normalized size of antiderivative = 1., 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 = {261} $-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2]

Rule 261

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*}

Mathematica [A]  time = 0.0004805, size = 13, normalized size = 1. $-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2]

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Maple [A]  time = 0.003, size = 17, normalized size = 1.3 \begin{align*}{ \left ( -1+x \right ) \left ( 1+x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.923715, size = 15, normalized size = 1.15 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}

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 = 1.81101, size = 23, normalized size = 1.77 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}

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 = 0.127718, size = 8, normalized size = 0.62 \begin{align*} - \sqrt{1 - x^{2}} \end{align*}

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 = 1.05616, size = 15, normalized size = 1.15 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}

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)