3.1548 \(\int \frac{1}{\sqrt{-2-b x} \sqrt{2-b x}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\cosh ^{-1}\left (-\frac{b x}{2}\right )}{b} \]

[Out]

-(ArcCosh[-(b*x)/2]/b)

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Rubi [A]  time = 0.003148, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {52} \[ -\frac{\cosh ^{-1}\left (-\frac{b x}{2}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-2 - b*x]*Sqrt[2 - b*x]),x]

[Out]

-(ArcCosh[-(b*x)/2]/b)

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2-b x} \sqrt{2-b x}} \, dx &=-\frac{\cosh ^{-1}\left (-\frac{b x}{2}\right )}{b}\\ \end{align*}

Mathematica [B]  time = 0.0045568, size = 27, normalized size = 2.25 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{-b x-2}}{\sqrt{2-b x}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-2 - b*x]*Sqrt[2 - b*x]),x]

[Out]

(-2*ArcTanh[Sqrt[-2 - b*x]/Sqrt[2 - b*x]])/b

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Maple [B]  time = 0.005, size = 61, normalized size = 5.1 \begin{align*}{\sqrt{ \left ( -bx-2 \right ) \left ( -bx+2 \right ) }\ln \left ({{b}^{2}x{\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}-4} \right ){\frac{1}{\sqrt{-bx-2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-b*x-2)*(-b*x+2))^(1/2)/(-b*x-2)^(1/2)/(-b*x+2)^(1/2)*ln(b^2*x/(b^2)^(1/2)+(b^2*x^2-4)^(1/2))/(b^2)^(1/2)

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Maxima [B]  time = 0.951835, size = 43, normalized size = 3.58 \begin{align*} \frac{\log \left (2 \, b^{2} x + 2 \, \sqrt{b^{2} x^{2} - 4} \sqrt{b^{2}}\right )}{\sqrt{b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*b^2*x + 2*sqrt(b^2*x^2 - 4)*sqrt(b^2))/sqrt(b^2)

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Fricas [B]  time = 2.01459, size = 62, normalized size = 5.17 \begin{align*} -\frac{\log \left (-b x + \sqrt{-b x + 2} \sqrt{-b x - 2}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-b*x + sqrt(-b*x + 2)*sqrt(-b*x - 2))/b

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Sympy [C]  time = 3.25108, size = 78, normalized size = 6.5 \begin{align*} - \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{4}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), 4/(b**2*x**2))/(4*pi**(3/2)*b) - I*me
ijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), 4*exp_polar(-2*I*pi)/(b**2*x**2))/(4
*pi**(3/2)*b)

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Giac [B]  time = 1.10542, size = 35, normalized size = 2.92 \begin{align*} \frac{2 \, \log \left ({\left | -\sqrt{-b x + 2} + \sqrt{-b x - 2} \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(abs(-sqrt(-b*x + 2) + sqrt(-b*x - 2)))/b