3.639 \(\int \frac{\sqrt{2+3 x}}{1-x^2} \, dx\)

Optimal. Leaf size=35 \[ \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{5}}\right )-\tan ^{-1}\left (\sqrt{3 x+2}\right ) \]

[Out]

-ArcTan[Sqrt[2 + 3*x]] + Sqrt[5]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[5]]

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Rubi [A]  time = 0.061711, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{5}}\right )-\tan ^{-1}\left (\sqrt{3 x+2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + 3*x]/(1 - x^2),x]

[Out]

-ArcTan[Sqrt[2 + 3*x]] + Sqrt[5]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[5]]

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Rubi in Sympy [A]  time = 10.6299, size = 31, normalized size = 0.89 \[ - \operatorname{atan}{\left (\sqrt{3 x + 2} \right )} + \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{3 x + 2}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(1/2)/(-x**2+1),x)

[Out]

-atan(sqrt(3*x + 2)) + sqrt(5)*atanh(sqrt(5)*sqrt(3*x + 2)/5)

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Mathematica [A]  time = 0.0386955, size = 35, normalized size = 1. \[ \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{3 x+2}}{\sqrt{5}}\right )-\tan ^{-1}\left (\sqrt{3 x+2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 + 3*x]/(1 - x^2),x]

[Out]

-ArcTan[Sqrt[2 + 3*x]] + Sqrt[5]*ArcTanh[Sqrt[2 + 3*x]/Sqrt[5]]

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Maple [A]  time = 0.018, size = 29, normalized size = 0.8 \[ -\arctan \left ( \sqrt{2+3\,x} \right ) +{\it Artanh} \left ({\frac{\sqrt{5}}{5}\sqrt{2+3\,x}} \right ) \sqrt{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(1/2)/(-x^2+1),x)

[Out]

-arctan((2+3*x)^(1/2))+arctanh(1/5*(2+3*x)^(1/2)*5^(1/2))*5^(1/2)

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Maxima [A]  time = 0.780385, size = 61, normalized size = 1.74 \[ -\frac{1}{2} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{3 \, x + 2}}{\sqrt{5} + \sqrt{3 \, x + 2}}\right ) - \arctan \left (\sqrt{3 \, x + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x + 2)/(x^2 - 1),x, algorithm="maxima")

[Out]

-1/2*sqrt(5)*log(-(sqrt(5) - sqrt(3*x + 2))/(sqrt(5) + sqrt(3*x + 2))) - arctan(
sqrt(3*x + 2))

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Fricas [A]  time = 0.220397, size = 54, normalized size = 1.54 \[ \frac{1}{2} \, \sqrt{5} \log \left (\frac{2 \, \sqrt{5} \sqrt{3 \, x + 2} + 3 \, x + 7}{x - 1}\right ) - \arctan \left (\sqrt{3 \, x + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x + 2)/(x^2 - 1),x, algorithm="fricas")

[Out]

1/2*sqrt(5)*log((2*sqrt(5)*sqrt(3*x + 2) + 3*x + 7)/(x - 1)) - arctan(sqrt(3*x +
2))

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Sympy [A]  time = 6.07904, size = 70, normalized size = 2. \[ - 5 \left (\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left (\frac{\sqrt{5} \sqrt{3 x + 2}}{5} \right )}}{5} & \text{for}\: 3 x + 2 > 5 \\- \frac{\sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{3 x + 2}}{5} \right )}}{5} & \text{for}\: 3 x + 2 < 5 \end{cases}\right ) - \operatorname{atan}{\left (\sqrt{3 x + 2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**(1/2)/(-x**2+1),x)

[Out]

-5*Piecewise((-sqrt(5)*acoth(sqrt(5)*sqrt(3*x + 2)/5)/5, 3*x + 2 > 5), (-sqrt(5)
*atanh(sqrt(5)*sqrt(3*x + 2)/5)/5, 3*x + 2 < 5)) - atan(sqrt(3*x + 2))

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GIAC/XCAS [A]  time = 0.213022, size = 65, normalized size = 1.86 \[ -\frac{1}{2} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{5} + 2 \, \sqrt{3 \, x + 2} \right |}}{2 \,{\left (\sqrt{5} + \sqrt{3 \, x + 2}\right )}}\right ) - \arctan \left (\sqrt{3 \, x + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x + 2)/(x^2 - 1),x, algorithm="giac")

[Out]

-1/2*sqrt(5)*ln(1/2*abs(-2*sqrt(5) + 2*sqrt(3*x + 2))/(sqrt(5) + sqrt(3*x + 2)))
- arctan(sqrt(3*x + 2))