3.1534 \(\int \frac{1}{\sqrt{-2+b x} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=11 \[ \frac{\cosh ^{-1}\left (\frac{b x}{2}\right )}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0022657, 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, 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 = 0.002413, size = 25, normalized size = 2.27 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x-2}}{\sqrt{b x+2}}\right )}{b} \]

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

________________________________________________________________________________________

Maple [B]  time = 0., size = 57, normalized size = 5.2 \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)

________________________________________________________________________________________

Maxima [B]  time = 0.98159, size = 43, normalized size = 3.91 \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)

________________________________________________________________________________________

Fricas [B]  time = 2.04784, size = 59, normalized size = 5.36 \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

________________________________________________________________________________________

Sympy [C]  time = 3.18175, size = 75, normalized size = 6.82 \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 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{4}{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*exp_polar(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)), 4/(b**2*x**2))/(4*p
i**(3/2)*b)

________________________________________________________________________________________

Giac [B]  time = 1.10346, size = 32, normalized size = 2.91 \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