Optimal. Leaf size=22 \[ x \left (-\sqrt{\frac{a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]
[Out]
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Rubi [A] time = 0.0249561, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ x \left (-\sqrt{\frac{a}{x^2}}\right ) \tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a/x^2]/Sqrt[1 + x^2],x]
[Out]
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Rubi in Sympy [A] time = 8.99198, size = 20, normalized size = 0.91 \[ - x \sqrt{\frac{a}{x^{2}}} \operatorname{atanh}{\left (\sqrt{x^{2} + 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a/x**2)**(1/2)/(x**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0122486, size = 28, normalized size = 1.27 \[ x \sqrt{\frac{a}{x^2}} \left (\log (x)-\log \left (\sqrt{x^2+1}+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a/x^2]/Sqrt[1 + x^2],x]
[Out]
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Maple [A] time = 0.008, size = 19, normalized size = 0.9 \[ -\sqrt{{\frac{a}{{x}^{2}}}}x{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{2}+1}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a/x^2)^(1/2)/(x^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 0.783211, size = 14, normalized size = 0.64 \[ -\sqrt{a} \operatorname{arsinh}\left (\frac{1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^2)/sqrt(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280684, size = 1, normalized size = 0.05 \[ \left [x \sqrt{\frac{a}{x^{2}}} \log \left (\frac{x^{2} - \sqrt{x^{2} + 1}{\left (x + 1\right )} + x + 1}{x^{2} - \sqrt{x^{2} + 1} x}\right ), -2 \, \sqrt{-a} \arctan \left (-\frac{a x - \sqrt{x^{2} + 1} a}{\sqrt{-a} x \sqrt{\frac{a}{x^{2}}}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^2)/sqrt(x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{2}}}}{\sqrt{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a/x**2)**(1/2)/(x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.266763, size = 41, normalized size = 1.86 \[ -\frac{1}{2} \, \sqrt{a}{\left ({\rm ln}\left (\sqrt{x^{2} + 1} + 1\right ) -{\rm ln}\left (\sqrt{x^{2} + 1} - 1\right )\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^2)/sqrt(x^2 + 1),x, algorithm="giac")
[Out]