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

Optimal. Leaf size=111 \[ \frac{\sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (x^2+2\right ) x}{3 \sqrt{x^4+x^2+1}}+\tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{6 \sqrt{x^4+x^2+1}} \]

[Out]

-(x*(2 + x^2))/(3*Sqrt[1 + x^2 + x^4]) + (x*Sqrt[1 + x^2 + x^4])/(3*(1 + x^2)) +
ArcTan[x/Sqrt[1 + x^2 + x^4]] + ((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*El
lipticE[2*ArcTan[x], 1/4])/(6*Sqrt[1 + x^2 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.435122, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ \frac{\sqrt{x^4+x^2+1} x}{3 \left (x^2+1\right )}-\frac{\left (x^2+2\right ) x}{3 \sqrt{x^4+x^2+1}}+\tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{6 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x^2)^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)) +
ArcTan[x/Sqrt[1 + x^2 + x^4]] + ((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*El
lipticE[2*ArcTan[x], 1/4])/(6*Sqrt[1 + x^2 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Mathematica [C]  time = 0.574707, size = 168, normalized size = 1.51 \[ \frac{-2 x \left (x^2+1\right ) \left (x^2+2\right )-\sqrt [3]{-1} \left (x^2+1\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left (\left (5 \sqrt [3]{-1}-1\right ) F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+12 \sqrt [3]{-1} \Pi \left (\sqrt [3]{-1};-i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )+3 x \left (x^4+x^2+1\right )}{6 \left (x^2+1\right ) \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x*(1 + x^2)*(2 + x^2) + 3*x*(1 + x^2 + x^4) - (-1)^(1/3)*(1 + x^2)*Sqrt[1 +
(-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1
)^(2/3)] + (-1 + 5*(-1)^(1/3))*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] +
12*(-1)^(1/3)*EllipticPi[(-1)^(1/3), (-I)*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/(
6*(1 + x^2)*Sqrt[1 + x^2 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.034, size = 419, normalized size = 3.8 \[{\frac{x}{2\,{x}^{2}+2}\sqrt{{x}^{4}+{x}^{2}+1}}-2\,{\frac{1/6\,{x}^{3}+x/3}{\sqrt{{x}^{4}+{x}^{2}+1}}}-{\frac{5}{3\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{2}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{2}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+2\,{\frac{\sqrt{1+1/2\,{x}^{2}-i/2{x}^{2}\sqrt{3}}\sqrt{1+1/2\,{x}^{2}+i/2{x}^{2}\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}{\it EllipticPi} \left ( \sqrt{-1/2+i/2\sqrt{3}}x,- \left ( -1/2+i/2\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-1/2-i/2\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*(x^4+x^2+1)^(1/2)/(x^2+1)-2*(1/6*x^3+1/3*x)/(x^4+x^2+1)^(1/2)-5/3/(-2+2*I*
3^(1/2))^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^2+1/2*I*x^2*3^(1/2))
^(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))+2/3/(-2+2*I*3^(1/2))^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/
2*x^2+1/2*I*x^2*3^(1/2))^(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))-2/3/(-2+2*I*3^(1/2))^(1/2)*(1+
1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^2+1/2*I*x^2*3^(1/2))^(1/2)/(x^4+x^2+1)
^(1/2)/(I*3^(1/2)+1)*EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))
^(1/2))+2/(-1/2+1/2*I*3^(1/2))^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*
x^2+1/2*I*x^2*3^(1/2))^(1/2)/(x^4+x^2+1)^(1/2)*EllipticPi((-1/2+1/2*I*3^(1/2))^(
1/2)*x,-1/(-1/2+1/2*I*3^(1/2)),(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I*3^(1/2))^(
1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{8} + 3 \, x^{6} + 4 \, x^{4} + 3 \, x^{2} + 1\right )} \sqrt{x^{4} + x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((x^8 + 3*x^6 + 4*x^4 + 3*x^2 + 1)*sqrt(x^4 + x^2 + 1)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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