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

Optimal. Leaf size=96 \[ -\frac{\sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac{\left (x^2+2\right ) x}{3 \sqrt{x^4+x^2+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.0585351, antiderivative size = 96, 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 \[ -\frac{\sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac{\left (x^2+2\right ) x}{3 \sqrt{x^4+x^2+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.466, size = 85, normalized size = 0.89 \[ \frac{x \left (x^{2} + 2\right )}{3 \sqrt{x^{4} + x^{2} + 1}} - \frac{x \sqrt{x^{4} + x^{2} + 1}}{3 \left (x^{2} + 1\right )} + \frac{\sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{3 \sqrt{x^{4} + x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.300801, size = 160, normalized size = 1.67 \[ \frac{x^3-\frac{1}{2} i \sqrt{2+\left (1+i \sqrt{3}\right ) x^2} \sqrt{6+\left (3-3 i \sqrt{3}\right ) x^2} F\left (\sin ^{-1}\left (\frac{1}{2} \left (i \sqrt{3} x+x\right )\right )|\frac{1}{2} i \left (i+\sqrt{3}\right )\right )-\sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+2 x}{3 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x + x^3 - (-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*Ellipt
icE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - (I/2)*Sqrt[2 + (1 + I*Sqrt[3])*x^2]*S
qrt[6 + (3 - (3*I)*Sqrt[3])*x^2]*EllipticF[ArcSin[(x + I*Sqrt[3]*x)/2], (I/2)*(I
+ Sqrt[3])])/(3*Sqrt[1 + x^2 + x^4])

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Maple [C]  time = 0.01, size = 247, normalized size = 2.6 \[ -2\,{\frac{-x/6+1/6\,{x}^{3}}{\sqrt{{x}^{4}+{x}^{2}+1}}}+{\frac{2}{3\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{4}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-2\,{\frac{-1/3\,{x}^{3}-x/6}{\sqrt{{x}^{4}+{x}^{2}+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-1/6*x+1/6*x^3)/(x^4+x^2+1)^(1/2)+2/3/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*
3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*Ellipti
cF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))+4/3/(-2+2*I*3^(1/2))
^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^
4+x^2+1)^(1/2)/(I*3^(1/2)+1)*(EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I
*3^(1/2))^(1/2))-EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/
2)))-2*(-1/3*x^3-1/6*x)/(x^4+x^2+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + 1}{\left (\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x**2 + 1)/((x**2 - x + 1)*(x**2 + x + 1))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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