3.9.54 \(\int \frac {1}{\sqrt {1-x}} \, dx\) [854]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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} \begin {gather*} -2 \sqrt {1-x} \end {gather*}

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 = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -2 \sqrt {1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - x],x]

[Out]

-2*Sqrt[1 - x]

________________________________________________________________________________________

Maple [A]
time = 0.55, size = 10, normalized size = 0.91

method	result	size
gosper	\(-2 \sqrt {1-x}\)	\(10\)
derivativedivides	\(-2 \sqrt {1-x}\)	\(10\)
default	\(-2 \sqrt {1-x}\)	\(10\)
trager	\(-2 \sqrt {1-x}\)	\(10\)
risch	\(\frac {-2+2 x}{\sqrt {1-x}}\)	\(13\)
meijerg	\(-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-x}}{\sqrt {\pi }}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 9, normalized size = 0.82 \begin {gather*} -2 \, \sqrt {-x + 1} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 9, normalized size = 0.82 \begin {gather*} -2 \, \sqrt {-x + 1} \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 8, normalized size = 0.73 \begin {gather*} - 2 \sqrt {1 - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(1 - x)

________________________________________________________________________________________

Giac [A]
time = 3.25, size = 9, normalized size = 0.82 \begin {gather*} -2 \, \sqrt {-x + 1} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.19, size = 9, normalized size = 0.82 \begin {gather*} -2\,\sqrt {1-x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - x)^(1/2),x)

[Out]

-2*(1 - x)^(1/2)

________________________________________________________________________________________