Optimal. Leaf size=22 \[ \cosh ^{-1}(x)-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
[Out]
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Rubi [A] time = 0.0290471, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \cosh ^{-1}(x)-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[-1 + x]*Sqrt[1 + x])/x^2,x]
[Out]
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Rubi in Sympy [A] time = 3.60591, size = 17, normalized size = 0.77 \[ \operatorname{acosh}{\left (x \right )} - \frac{\sqrt{x - 1} \sqrt{x + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+x)**(1/2)*(1+x)**(1/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0210917, size = 38, normalized size = 1.73 \[ \log \left (x+\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[-1 + x]*Sqrt[1 + x])/x^2,x]
[Out]
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Maple [B] time = 0.01, size = 44, normalized size = 2. \[{\frac{1}{x}\sqrt{-1+x}\sqrt{1+x} \left ( \ln \left ( x+\sqrt{{x}^{2}-1} \right ) x-\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+x)^(1/2)*(1+x)^(1/2)/x^2,x)
[Out]
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Maxima [A] time = 1.52359, size = 36, normalized size = 1.64 \[ -\frac{\sqrt{x^{2} - 1}}{x} + \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]
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 = 0.22512, size = 80, normalized size = 3.64 \[ -\frac{{\left (\sqrt{x + 1} \sqrt{x - 1} x - x^{2}\right )} \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) - 1}{\sqrt{x + 1} \sqrt{x - 1} x - x^{2}} \]
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. \[ \int \frac{\sqrt{x - 1} \sqrt{x + 1}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-1+x)**(1/2)*(1+x)**(1/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.228453, size = 54, normalized size = 2.45 \[ -\frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} - \frac{1}{2} \,{\rm ln}\left ({\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + 1)*sqrt(x - 1)/x^2,x, algorithm="giac")
[Out]