3.1550 \(\int \frac{1}{\sqrt{-4+b x} \sqrt{4+b x}} \, dx\)

Optimal. Leaf size=11 \[ \frac{\cosh ^{-1}\left (\frac{b x}{4}\right )}{b} \]

[Out]

ArcCosh[(b*x)/4]/b

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Rubi [A]  time = 0.0178026, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\cosh ^{-1}\left (\frac{b x}{4}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-4 + b*x]*Sqrt[4 + b*x]),x]

[Out]

ArcCosh[(b*x)/4]/b

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Rubi in Sympy [A]  time = 4.193, size = 7, normalized size = 0.64 \[ \frac{\operatorname{acosh}{\left (\frac{b x}{4} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x-4)**(1/2)/(b*x+4)**(1/2),x)

[Out]

acosh(b*x/4)/b

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Mathematica [B]  time = 0.0120525, size = 24, normalized size = 2.18 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x-4}}{2 \sqrt{2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-4 + b*x]*Sqrt[4 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[-4 + b*x]/(2*Sqrt[2])])/b

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Maple [B]  time = 0.01, size = 57, normalized size = 5.2 \[{1\sqrt{ \left ( bx-4 \right ) \left ( bx+4 \right ) }\ln \left ({{b}^{2}x{\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}-16} \right ){\frac{1}{\sqrt{bx-4}}}{\frac{1}{\sqrt{bx+4}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x-4)^(1/2)/(b*x+4)^(1/2),x)

[Out]

((b*x-4)*(b*x+4))^(1/2)/(b*x-4)^(1/2)/(b*x+4)^(1/2)*ln(b^2*x/(b^2)^(1/2)+(b^2*x^
2-16)^(1/2))/(b^2)^(1/2)

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Maxima [A]  time = 1.34354, size = 43, normalized size = 3.91 \[ \frac{\log \left (2 \, b^{2} x + 2 \, \sqrt{b^{2} x^{2} - 16} \sqrt{b^{2}}\right )}{\sqrt{b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 4)*sqrt(b*x - 4)),x, algorithm="maxima")

[Out]

log(2*b^2*x + 2*sqrt(b^2*x^2 - 16)*sqrt(b^2))/sqrt(b^2)

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Fricas [A]  time = 0.202789, size = 35, normalized size = 3.18 \[ -\frac{\log \left (-b x + \sqrt{b x + 4} \sqrt{b x - 4}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 4)*sqrt(b*x - 4)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.80517, size = 75, normalized size = 6.82 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{16 e^{2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{16}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x-4)**(1/2)/(b*x+4)**(1/2),x)

[Out]

meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), 16*exp_p
olar(2*I*pi)/(b**2*x**2))/(4*pi**(3/2)*b) + I*meijerg(((-1/2, -1/4, 0, 1/4, 1/2,
1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), 16/(b**2*x**2))/(4*pi**(3/2)*b)

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GIAC/XCAS [A]  time = 0.241617, size = 32, normalized size = 2.91 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 4} + \sqrt{b x - 4} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 4)*sqrt(b*x - 4)),x, algorithm="giac")

[Out]

-2*ln(abs(-sqrt(b*x + 4) + sqrt(b*x - 4)))/b