Optimal. Leaf size=9 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]
[Out]
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Rubi [A] time = 0.0139321, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 + x^(-2)]*x*(1 + x^2)),x]
[Out]
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Rubi in Sympy [A] time = 1.44591, 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(1/x/(x**2+1)/(1+1/x**2)**(1/2),x)
[Out]
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Mathematica [B] time = 0.00927791, size = 20, normalized size = 2.22 \[ \frac{\sqrt{\frac{1}{x^2}+1} x^2}{x^2+1} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 + x^(-2)]*x*(1 + x^2)),x]
[Out]
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Maple [A] time = 0.005, size = 12, normalized size = 1.3 \[{\frac{1}{\sqrt{{\frac{{x}^{2}+1}{{x}^{2}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(x^2+1)/(1+1/x^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.813992, size = 12, normalized size = 1.33 \[ \frac{x}{\sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 1)*x*sqrt(1/x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260374, 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(1/((x^2 + 1)*x*sqrt(1/x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.91692, 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] integrate(1/x/(x**2+1)/(1+1/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 1\right )} x \sqrt{\frac{1}{x^{2}} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 1)*x*sqrt(1/x^2 + 1)),x, algorithm="giac")
[Out]