3.917 \(\int \frac{1}{\sqrt{1-a x} (1+a x)} \, dx\)

Optimal. Leaf size=27 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]

[Out]

-((Sqrt[2]*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/a)

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Rubi [A]  time = 0.0375804, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2]*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/a)

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Rubi in Sympy [A]  time = 4.9382, size = 24, normalized size = 0.89 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{- a x + 1}}{2} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*atanh(sqrt(2)*sqrt(-a*x + 1)/2)/a

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Mathematica [A]  time = 0.0166279, size = 27, normalized size = 1. \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2]*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/a)

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Maple [A]  time = 0.008, size = 23, normalized size = 0.9 \[ -{\frac{\sqrt{2}}{a}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-ax+1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x+1)/(-a*x+1)^(1/2),x)

[Out]

-arctanh(1/2*(-a*x+1)^(1/2)*2^(1/2))*2^(1/2)/a

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Maxima [A]  time = 0.785528, size = 55, normalized size = 2.04 \[ \frac{\sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \sqrt{-a x + 1}\right )}}{2 \, \sqrt{2} + 2 \, \sqrt{-a x + 1}}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*log(-2*(sqrt(2) - sqrt(-a*x + 1))/((2*sqrt(2)) + 2*sqrt(-a*x + 1)))/
a

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Fricas [A]  time = 0.237862, size = 47, normalized size = 1.74 \[ \frac{\sqrt{2} \log \left (\frac{a x + 2 \, \sqrt{2} \sqrt{-a x + 1} - 3}{a x + 1}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*log((a*x + 2*sqrt(2)*sqrt(-a*x + 1) - 3)/(a*x + 1))/a

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Sympy [A]  time = 7.00468, size = 65, normalized size = 2.41 \[ \begin{cases} \frac{2 \left (\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left (\frac{\sqrt{2}}{\sqrt{- a x + 1}} \right )}}{2} & \text{for}\: \frac{1}{- a x + 1} > \frac{1}{2} \\- \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2}}{\sqrt{- a x + 1}} \right )}}{2} & \text{for}\: \frac{1}{- a x + 1} < \frac{1}{2} \end{cases}\right )}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*Piecewise((-sqrt(2)*acoth(sqrt(2)/sqrt(-a*x + 1))/2, 1/(-a*x + 1) >
1/2), (-sqrt(2)*atanh(sqrt(2)/sqrt(-a*x + 1))/2, 1/(-a*x + 1) < 1/2))/a, Ne(a,
0)), (x, True))

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GIAC/XCAS [A]  time = 0.211812, size = 57, normalized size = 2.11 \[ \frac{\sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{-a x + 1} \right |}}{2 \,{\left (\sqrt{2} + \sqrt{-a x + 1}\right )}}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*ln(1/2*abs(-2*sqrt(2) + 2*sqrt(-a*x + 1))/(sqrt(2) + sqrt(-a*x + 1))
)/a