3.9 \(\int \frac{2+x^2}{\left (1+x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx\)

Optimal. Leaf size=48 \[ \frac{\sqrt{2} \left (x^2+2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

[Out]

(Sqrt[2]*(2 + x^2)*EllipticE[ArcTan[x], 1/2])/(Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2
+ 3*x^2 + x^4])

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Rubi [A]  time = 0.099812, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\sqrt{2} \left (x^2+2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + x^2)/((1 + x^2)*Sqrt[2 + 3*x^2 + x^4]),x]

[Out]

(Sqrt[2]*(2 + x^2)*EllipticE[ArcTan[x], 1/2])/(Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2
+ 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 7.00978, size = 42, normalized size = 0.88 \[ \frac{\sqrt{2} \sqrt{x^{4} + 3 x^{2} + 2} E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{\sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+2)/(x**2+1)/(x**4+3*x**2+2)**(1/2),x)

[Out]

sqrt(2)*sqrt(x**4 + 3*x**2 + 2)*elliptic_e(atan(x), 1/2)/(sqrt((x**2 + 2)/(x**2
+ 1))*(x**2 + 1))

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Mathematica [C]  time = 0.100207, size = 94, normalized size = 1.96 \[ \frac{x^3-i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+2 x}{\sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x^2)/((1 + x^2)*Sqrt[2 + 3*x^2 + x^4]),x]

[Out]

(2*x + x^3 + I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] -
I*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/Sqrt[2 + 3*x^2
+ x^4]

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Maple [C]  time = 0.021, size = 81, normalized size = 1.7 \[{ \left ({x}^{2}+2 \right ) x{\frac{1}{\sqrt{ \left ({x}^{2}+1 \right ) \left ({x}^{2}+2 \right ) }}}}-{{\frac{i}{2}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2)/(x^2+1)/(x^4+3*x^2+2)^(1/2),x)

[Out]

(x^2+2)*x/((x^2+1)*(x^2+2))^(1/2)-1/2*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x
^4+3*x^2+2)^(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2*I*2^(1/2)*x,
2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 2}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2)/(sqrt(x^4 + 3*x^2 + 2)*(x^2 + 1)),x, algorithm="maxima")

[Out]

integrate((x^2 + 2)/(sqrt(x^4 + 3*x^2 + 2)*(x^2 + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2} + 2}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (x^{2} + 1\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2)/(sqrt(x^4 + 3*x^2 + 2)*(x^2 + 1)),x, algorithm="fricas")

[Out]

integral((x^2 + 2)/(sqrt(x^4 + 3*x^2 + 2)*(x^2 + 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 2}{\sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (x^{2} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+2)/(x**2+1)/(x**4+3*x**2+2)**(1/2),x)

[Out]

Integral((x**2 + 2)/(sqrt((x**2 + 1)*(x**2 + 2))*(x**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 2}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 2)/(sqrt(x^4 + 3*x^2 + 2)*(x^2 + 1)),x, algorithm="giac")

[Out]

integrate((x^2 + 2)/(sqrt(x^4 + 3*x^2 + 2)*(x^2 + 1)), x)