Optimal. Leaf size=9 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]
[Out]
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Rubi [A] time = 0.0140857, antiderivative size = 9, 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 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 + x^(-2)]*x)/(1 + x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 1.41876, size = 10, normalized size = 1.11 \[ \frac{1}{\sqrt{1 + \frac{1}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(1+1/x**2)**(1/2)/(x**2+1)**2,x)
[Out]
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Mathematica [B] time = 0.0128918, size = 20, normalized size = 2.22 \[ \frac{\sqrt{\frac{1}{x^2}+1} x^2}{x^2+1} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 + x^(-2)]*x)/(1 + x^2)^2,x]
[Out]
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Maple [B] time = 0.005, size = 23, normalized size = 2.6 \[{\frac{{x}^{2}}{{x}^{2}+1}\sqrt{{\frac{{x}^{2}+1}{{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(1+1/x^2)^(1/2)/(x^2+1)^2,x)
[Out]
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Maxima [A] time = 0.804031, size = 15, normalized size = 1.67 \[ \frac{1}{\sqrt{\frac{x^{2} + 1}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sqrt(1/x^2 + 1)/(x^2 + 1)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.261212, size = 38, normalized size = 4.22 \[ \frac{x^{2} \sqrt{\frac{x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sqrt(1/x^2 + 1)/(x^2 + 1)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.11112, size = 8, normalized size = 0.89 \[ \frac{x}{\sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(1+1/x**2)**(1/2)/(x**2+1)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.264072, size = 15, normalized size = 1.67 \[ \frac{x{\rm sign}\left (x\right )}{\sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sqrt(1/x^2 + 1)/(x^2 + 1)^2,x, algorithm="giac")
[Out]