3.854 \(\int \frac{1}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=11 \[ -2 \sqrt{1-x} \]

[Out]

-2*Sqrt[1 - x]

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Rubi [A]  time = 0.0009113, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ -2 \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - x],x]

[Out]

-2*Sqrt[1 - x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0027051, size = 11, normalized size = 1. \[ -2 \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - x],x]

[Out]

-2*Sqrt[1 - x]

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Maple [A]  time = 0., size = 10, normalized size = 0.9 \begin{align*} -2\,\sqrt{1-x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.40839, size = 12, normalized size = 1.09 \begin{align*} -2 \, \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-x + 1)

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Fricas [A]  time = 1.4112, size = 23, normalized size = 2.09 \begin{align*} -2 \, \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-x + 1)

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Sympy [A]  time = 0.052253, size = 8, normalized size = 0.73 \begin{align*} - 2 \sqrt{1 - x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(1 - x)

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Giac [A]  time = 1.14489, size = 12, normalized size = 1.09 \begin{align*} -2 \, \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-x + 1)