Optimal. Leaf size=20 \[ \frac{\sqrt{x^2} \sqrt{x^2+1}}{x} \]
[Out]
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Rubi [A] time = 0.0108797, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{x^2} \sqrt{x^2+1}}{x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x^2/(1 + x^2)],x]
[Out]
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Rubi in Sympy [A] time = 3.40483, size = 15, normalized size = 0.75 \[ \frac{\sqrt{x^{2} + 1} \sqrt{x^{2}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2/(x**2+1))**(1/2),x)
[Out]
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Mathematica [A] time = 0.00926511, size = 17, normalized size = 0.85 \[ \frac{x}{\sqrt{\frac{x^2}{x^2+1}}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x^2/(1 + x^2)],x]
[Out]
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Maple [A] time = 0.005, size = 23, normalized size = 1.2 \[{\frac{{x}^{2}+1}{x}\sqrt{{\frac{{x}^{2}}{{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2/(x^2+1))^(1/2),x)
[Out]
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Maxima [A] time = 0.757597, size = 9, normalized size = 0.45 \[ \sqrt{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2/(x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.262364, size = 30, normalized size = 1.5 \[ \frac{{\left (x^{2} + 1\right )} \sqrt{\frac{x^{2}}{x^{2} + 1}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2/(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.8012, size = 36, normalized size = 1.8 \[ x \sqrt{x^{2}} \sqrt{\frac{1}{x^{2} + 1}} + \frac{\sqrt{x^{2}} \sqrt{\frac{1}{x^{2} + 1}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2/(x**2+1))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.258509, size = 20, normalized size = 1. \[ \sqrt{x^{2} + 1}{\rm sign}\left (x\right ) -{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2/(x^2 + 1)),x, algorithm="giac")
[Out]