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

Optimal. Leaf size=137 \[ \frac{4 \sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac{1}{3} \sqrt{x^4+x^2+1} x+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{\sqrt{x^4+x^2+1}}-\frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.110453, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{4 \sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac{1}{3} \sqrt{x^4+x^2+1} x+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{\sqrt{x^4+x^2+1}}-\frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.9895, size = 128, normalized size = 0.93 \[ \frac{x \sqrt{x^{4} + x^{2} + 1}}{3} + \frac{4 x \sqrt{x^{4} + x^{2} + 1}}{3 \left (x^{2} + 1\right )} - \frac{4 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{3 \sqrt{x^{4} + x^{2} + 1}} + \frac{\sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{\sqrt{x^{4} + x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**4 + x**2 + 1)/3 + 4*x*sqrt(x**4 + x**2 + 1)/(3*(x**2 + 1)) - 4*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)) + sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_f(2*a
tan(x), 1/4)/sqrt(x**4 + x**2 + 1)

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Mathematica [C]  time = 0.193832, size = 143, normalized size = 1.04 \[ \frac{x^5+x^3+2 \sqrt [3]{-1} \left (\sqrt [3]{-1}-2\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+4 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x}{3 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x + x^3 + x^5 + 4*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*
EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 2*(-1)^(1/3)*(-2 + (-1)^(1/3))*
Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSinh[(-1)^(5/6)
*x], (-1)^(2/3)])/(3*Sqrt[1 + x^2 + x^4])

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Maple [C]  time = 0.01, size = 218, normalized size = 1.6 \[{\frac{4}{3\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{x}{3}\sqrt{{x}^{4}+{x}^{2}+1}}-{\frac{16}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)*EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-
2+2*I*3^(1/2))^(1/2))+1/3*x*(x^4+x^2+1)^(1/2)-16/3/(-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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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