Optimal. Leaf size=36 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
[Out]
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Rubi [A] time = 0.0438559, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-1 + x]/(x^2*Sqrt[1 + x]),x]
[Out]
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Rubi in Sympy [A] time = 2.1202, size = 29, normalized size = 0.81 \[ \operatorname{atan}{\left (\sqrt{x - 1} \sqrt{x + 1} \right )} - \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+x)**(1/2)/x**2/(1+x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0458833, size = 62, normalized size = 1.72 \[ -\frac{\sqrt{\frac{x-1}{x+1}} \left (\sqrt{x-1} (x+1)+x \sqrt{x+1} \tan ^{-1}\left (\frac{1}{\sqrt{x-1} \sqrt{x+1}}\right )\right )}{\sqrt{x-1} x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-1 + x]/(x^2*Sqrt[1 + x]),x]
[Out]
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Maple [A] time = 0.022, size = 43, normalized size = 1.2 \[{\frac{1}{x} \left ( -\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) x-\sqrt{{x}^{2}-1} \right ) \sqrt{-1+x}\sqrt{1+x}{\frac{1}{\sqrt{{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+x)^(1/2)/x^2/(1+x)^(1/2),x)
[Out]
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Maxima [A] time = 0.79807, size = 27, normalized size = 0.75 \[ -\frac{\sqrt{x^{2} - 1}}{x} - \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(sqrt(x + 1)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.295791, size = 80, normalized size = 2.22 \[ \frac{2 \,{\left (\sqrt{x + 1} \sqrt{x - 1} x - x^{2}\right )} \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) + 1}{\sqrt{x + 1} \sqrt{x - 1} x - x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(sqrt(x + 1)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x - 1}}{x^{2} \sqrt{x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-1+x)**(1/2)/x**2/(1+x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.271331, size = 57, normalized size = 1.58 \[ -\frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} - 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 1)/(sqrt(x + 1)*x^2),x, algorithm="giac")
[Out]