\(\int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx\) [1534]

Optimal result

Integrand size = 19, antiderivative size = 11 \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=\frac {\text {arccosh}\left (\frac {b x}{2}\right )}{b} \]

[Out]

arccosh(1/2*b*x)/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {54} \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=\frac {\text {arccosh}\left (\frac {b x}{2}\right )}{b} \]

[In]

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

[Out]

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

Rule 54

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*} \text {integral}& = \frac {\cosh ^{-1}\left (\frac {b x}{2}\right )}{b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(11)=22\).

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.27 \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {2+b x}}{\sqrt {-2+b x}}\right )}{b} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(9)=18\).

Time = 0.51 (sec) , antiderivative size = 57, normalized size of antiderivative = 5.18

method	result	size
default	\(\frac {\sqrt {\left (b x -2\right ) \left (b x +2\right )}\, \ln \left (\frac {b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-4}\right )}{\sqrt {b x -2}\, \sqrt {b x +2}\, \sqrt {b^{2}}}\)	\(57\)

[In]

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (9) = 18\).

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=-\frac {\log \left (-b x + \sqrt {b x + 2} \sqrt {b x - 2}\right )}{b} \]

[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] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.90 (sec) , antiderivative size = 75, normalized size of antiderivative = 6.82 \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=\frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {4 e^{2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {4}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (9) = 18\).

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=\frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 4} b\right )}{b} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (9) = 18\).

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=-\frac {2 \, \log \left (\sqrt {b x + 2} - \sqrt {b x - 2}\right )}{b} \]

[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

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 4.55 \[ \int \frac {1}{\sqrt {-2+b x} \sqrt {2+b x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {b\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]

[In]

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

[Out]

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