3.240 \(\int \frac{x}{\left (3-x^2\right ) \sqrt{5-x^2}} \, dx\)

Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{5-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

ArcTanh[Sqrt[5 - x^2]/Sqrt[2]]/Sqrt[2]

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Rubi [A]  time = 0.0588942, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{5-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[x/((3 - x^2)*Sqrt[5 - x^2]),x]

[Out]

ArcTanh[Sqrt[5 - x^2]/Sqrt[2]]/Sqrt[2]

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Rubi in Sympy [A]  time = 4.42922, size = 22, normalized size = 0.92 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{- x^{2} + 5}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**2+3)/(-x**2+5)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.017568, size = 24, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{5-x^2}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/((3 - x^2)*Sqrt[5 - x^2]),x]

[Out]

ArcTanh[Sqrt[5 - x^2]/Sqrt[2]]/Sqrt[2]

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Maple [B]  time = 0.047, size = 100, normalized size = 4.2 \[{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{ \left ( 4-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +2}}}} \right ) }+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{ \left ( 4+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^2+3)/(-x^2+5)^(1/2),x)

[Out]

1/4*2^(1/2)*arctanh(1/4*(4-2*3^(1/2)*(x-3^(1/2)))*2^(1/2)/(-(x-3^(1/2))^2-2*3^(1
/2)*(x-3^(1/2))+2)^(1/2))+1/4*2^(1/2)*arctanh(1/4*(4+2*3^(1/2)*(x+3^(1/2)))*2^(1
/2)/(-(x+3^(1/2))^2+2*3^(1/2)*(x+3^(1/2))+2)^(1/2))

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Maxima [A]  time = 1.52851, size = 151, normalized size = 6.29 \[ \frac{1}{12} \, \sqrt{3}{\left (\sqrt{3} \sqrt{2} \log \left (\sqrt{3} + \frac{2 \, \sqrt{2} \sqrt{-x^{2} + 5}}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}} + \frac{4}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}}\right ) + \sqrt{3} \sqrt{2} \log \left (-\sqrt{3} + \frac{2 \, \sqrt{2} \sqrt{-x^{2} + 5}}{{\left | 2 \, x - 2 \, \sqrt{3} \right |}} + \frac{4}{{\left | 2 \, x - 2 \, \sqrt{3} \right |}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^2 - 3)*sqrt(-x^2 + 5)),x, algorithm="maxima")

[Out]

1/12*sqrt(3)*(sqrt(3)*sqrt(2)*log(sqrt(3) + 2*sqrt(2)*sqrt(-x^2 + 5)/abs(2*x + 2
*sqrt(3)) + 4/abs(2*x + 2*sqrt(3))) + sqrt(3)*sqrt(2)*log(-sqrt(3) + 2*sqrt(2)*s
qrt(-x^2 + 5)/abs(2*x - 2*sqrt(3)) + 4/abs(2*x - 2*sqrt(3))))

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Fricas [A]  time = 0.222103, size = 68, normalized size = 2.83 \[ \frac{1}{8} \, \sqrt{2} \log \left (\frac{\sqrt{2}{\left (x^{4} - 22 \, x^{2} + 89\right )} - 8 \,{\left (x^{2} - 7\right )} \sqrt{-x^{2} + 5}}{x^{4} - 6 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^2 - 3)*sqrt(-x^2 + 5)),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log((sqrt(2)*(x^4 - 22*x^2 + 89) - 8*(x^2 - 7)*sqrt(-x^2 + 5))/(x^4
- 6*x^2 + 9))

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Sympy [A]  time = 12.5044, size = 61, normalized size = 2.54 \[ - \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left (\frac{\sqrt{2}}{\sqrt{- x^{2} + 5}} \right )}}{2} & \text{for}\: \frac{1}{- x^{2} + 5} > \frac{1}{2} \\- \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2}}{\sqrt{- x^{2} + 5}} \right )}}{2} & \text{for}\: \frac{1}{- x^{2} + 5} < \frac{1}{2} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**2+3)/(-x**2+5)**(1/2),x)

[Out]

-Piecewise((-sqrt(2)*acoth(sqrt(2)/sqrt(-x**2 + 5))/2, 1/(-x**2 + 5) > 1/2), (-s
qrt(2)*atanh(sqrt(2)/sqrt(-x**2 + 5))/2, 1/(-x**2 + 5) < 1/2))

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GIAC/XCAS [A]  time = 0.21188, size = 57, normalized size = 2.38 \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} + \sqrt{-x^{2} + 5}\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{2} + \sqrt{-x^{2} + 5} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/((x^2 - 3)*sqrt(-x^2 + 5)),x, algorithm="giac")

[Out]

1/4*sqrt(2)*ln(sqrt(2) + sqrt(-x^2 + 5)) - 1/4*sqrt(2)*ln(abs(-sqrt(2) + sqrt(-x
^2 + 5)))