Optimal. Leaf size=47 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]
[Out]
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Rubi [A] time = 0.0285857, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 + x^2]*Sqrt[2 + x^2]),x]
[Out]
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Rubi in Sympy [A] time = 4.87794, size = 42, normalized size = 0.89 \[ \frac{\sqrt{2} \sqrt{x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**2+1)**(1/2)/(x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0288724, size = 19, normalized size = 0.4 \[ -\frac{i F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + x^2]),x]
[Out]
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Maple [C] time = 0.034, size = 15, normalized size = 0.3 \[ -i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^2+1)^(1/2)/(x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 1} \sqrt{x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**2+1)**(1/2)/(x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="giac")
[Out]