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

Optimal. Leaf size=32 \[ -\frac{\tanh ^{-1}\left (\frac{x+4}{2 \sqrt{2} \sqrt{-x^2+x+2}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTanh[(4 + x)/(2*Sqrt[2]*Sqrt[2 + x - x^2])]/Sqrt[2])

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Rubi [A]  time = 0.0331915, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{x+4}{2 \sqrt{2} \sqrt{-x^2+x+2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[2 + x - x^2]),x]

[Out]

-(ArcTanh[(4 + x)/(2*Sqrt[2]*Sqrt[2 + x - x^2])]/Sqrt[2])

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Rubi in Sympy [A]  time = 2.08042, size = 29, normalized size = 0.91 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (x + 4\right )}{4 \sqrt{- x^{2} + x + 2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*atanh(sqrt(2)*(x + 4)/(4*sqrt(-x**2 + x + 2)))/2

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Mathematica [A]  time = 0.0213003, size = 34, normalized size = 1.06 \[ \frac{\log (x)-\log \left (2 \sqrt{2} \sqrt{-x^2+x+2}+x+4\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(Log[x] - Log[4 + x + 2*Sqrt[2]*Sqrt[2 + x - x^2]])/Sqrt[2]

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Maple [A]  time = 0.006, size = 25, normalized size = 0.8 \[ -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{-{x}^{2}+x+2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^2+x+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.50994, size = 45, normalized size = 1.41 \[ -\frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{-x^{2} + x + 2}}{{\left | x \right |}} + \frac{4}{{\left | x \right |}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244018, size = 53, normalized size = 1.66 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{4 \, \sqrt{2} \sqrt{-x^{2} + x + 2}{\left (x + 4\right )} + 7 \, x^{2} - 16 \, x - 32}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*log(-(4*sqrt(2)*sqrt(-x^2 + x + 2)*(x + 4) + 7*x^2 - 16*x - 32)/x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{- \left (x - 2\right ) \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(-(x - 2)*(x + 1))), x)

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GIAC/XCAS [A]  time = 0.231367, size = 96, normalized size = 3. \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{2} + \frac{2 \,{\left (2 \, \sqrt{-x^{2} + x + 2} - 3\right )}}{2 \, x - 1} - 6 \right |}}{{\left | 4 \, \sqrt{2} + \frac{2 \,{\left (2 \, \sqrt{-x^{2} + x + 2} - 3\right )}}{2 \, x - 1} - 6 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*ln(abs(-4*sqrt(2) + 2*(2*sqrt(-x^2 + x + 2) - 3)/(2*x - 1) - 6)/abs
(4*sqrt(2) + 2*(2*sqrt(-x^2 + x + 2) - 3)/(2*x - 1) - 6))