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

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\cosh ^{-1}\left (-\frac {b x}{2}\right )}{b} \end {gather*}

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*}

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Mathematica [B]  time = 0.00, size = 27, normalized size = 2.25 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-b x-2}}{\sqrt {2-b x}}\right )}{b} \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.05, size = 27, normalized size = 2.25 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2-b x}}{\sqrt {-b x-2}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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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fricas [B]  time = 1.02, size = 28, normalized size = 2.33 \begin {gather*} -\frac {\log \left (-b x + \sqrt {-b x + 2} \sqrt {-b x - 2}\right )}{b} \end {gather*}

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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giac [B]  time = 1.10, size = 25, normalized size = 2.08 \begin {gather*} \frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x - 2}\right )}{b} \end {gather*}

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(sqrt(-b*x + 2) - sqrt(-b*x - 2))/b

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maple [B]  time = 0.01, size = 61, normalized size = 5.08 \begin {gather*} \frac {\sqrt {\left (-b x -2\right ) \left (-b x +2\right )}\, \ln \left (\frac {b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-4}\right )}{\sqrt {-b x -2}\, \sqrt {-b x +2}\, \sqrt {b^{2}}} \end {gather*}

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

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maxima [B]  time = 1.27, size = 26, normalized size = 2.17 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 4} b\right )}{b} \end {gather*}

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)*b)/b

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mupad [B]  time = 0.29, size = 52, normalized size = 4.33 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {-b\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {2-b\,x}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*atan((b*(2^(1/2)*1i - (- b*x - 2)^(1/2)))/((2^(1/2) - (2 - b*x)^(1/2))*(-b^2)^(1/2))))/(-b^2)^(1/2)

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sympy [C]  time = 4.63, size = 78, normalized size = 6.50 \begin {gather*} - \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 {gather*}

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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