3.1537 \(\int \frac{1}{\sqrt{2-b x} \sqrt{2+b x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0212299, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\sin ^{-1}\left (\frac{b x}{2}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.31366, size = 7, normalized size = 0.64 \[ \frac{\operatorname{asin}{\left (\frac{b x}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(b*x/2)/b

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Mathematica [A]  time = 0.0118723, size = 11, normalized size = 1. \[ \frac{\sin ^{-1}\left (\frac{b x}{2}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 56, normalized size = 5.1 \[{1\sqrt{ \left ( -bx+2 \right ) \left ( bx+2 \right ) }\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+4}}}} \right ){\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

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

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Maxima [A]  time = 1.54097, size = 24, normalized size = 2.18 \[ \frac{\arcsin \left (\frac{b^{2} x}{2 \, \sqrt{b^{2}}}\right )}{\sqrt{b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)),x, algorithm="maxima")

[Out]

arcsin(1/2*b^2*x/sqrt(b^2))/sqrt(b^2)

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Fricas [A]  time = 0.210135, size = 42, normalized size = 3.82 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b x + 2} - 2}{b x}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.84782, size = 76, normalized size = 6.91 \[ - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{4}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} \]

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]

-I*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) + meijerg(((-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/XCAS [A]  time = 0.215587, size = 20, normalized size = 1.82 \[ \frac{2 \, \arcsin \left (\frac{1}{2} \, \sqrt{b x + 2}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x + 2)),x, algorithm="giac")

[Out]

2*arcsin(1/2*sqrt(b*x + 2))/b