Optimal. Leaf size=19 \[ \tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right ) \]
[Out]
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Rubi [A] time = 0.0351389, antiderivative size = 19, 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 \[ \tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((1 - x)*Sqrt[-4 + 2*x + x^2]),x]
[Out]
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Rubi in Sympy [A] time = 6.13359, size = 19, normalized size = 1. \[ \operatorname{atan}{\left (\frac{- 4 x + 6}{2 \sqrt{x^{2} + 2 x - 4}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-x)/(x**2+2*x-4)**(1/2),x)
[Out]
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Mathematica [A] time = 0.026845, size = 31, normalized size = 1.63 \[ \tan ^{-1}\left (\frac{2 (1-x)+1}{\sqrt{(1-x)^2-4 (1-x)-1}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - x)*Sqrt[-4 + 2*x + x^2]),x]
[Out]
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Maple [A] time = 0.01, size = 23, normalized size = 1.2 \[ -\arctan \left ({\frac{-6+4\,x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+4\,x-5}}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-x)/(x^2+2*x-4)^(1/2),x)
[Out]
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Maxima [A] time = 0.744551, size = 36, normalized size = 1.89 \[ -\arcsin \left (\frac{2 \, \sqrt{5} x}{5 \,{\left | x - 1 \right |}} - \frac{3 \, \sqrt{5}}{5 \,{\left | x - 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(x^2 + 2*x - 4)*(x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21944, size = 24, normalized size = 1.26 \[ -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(x^2 + 2*x - 4)*(x - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{x \sqrt{x^{2} + 2 x - 4} - \sqrt{x^{2} + 2 x - 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-x)/(x**2+2*x-4)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.207849, size = 24, normalized size = 1.26 \[ -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(sqrt(x^2 + 2*x - 4)*(x - 1)),x, algorithm="giac")
[Out]