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

Optimal. Leaf size=118 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{x^4+x^2+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{2 \sqrt{x^4+x^2+1}} \]

[Out]

ArcTan[x/Sqrt[1 + x^2 + x^4]]/2 + ((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*E
llipticE[2*ArcTan[x], 1/4])/(2*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])/(4*Sqrt[1 + x^2 + x^4])

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Rubi [A]  time = 0.197711, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right )-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{4 \sqrt{x^4+x^2+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{2 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x/Sqrt[1 + x^2 + x^4]]/2 + ((1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*E
llipticE[2*ArcTan[x], 1/4])/(2*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])/(4*Sqrt[1 + x^2 + x^4])

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Rubi in Sympy [A]  time = 80.0594, size = 328, normalized size = 2.78 \[ \frac{\sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{2 \sqrt{x^{4} + x^{2} + 1}} - \frac{\sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{2 \sqrt{x^{4} + x^{2} + 1}} - \frac{4 i \sqrt{x^{4} + x^{2} + 1} \Im{\left (\frac{1}{\left (1 + \sqrt{3} i\right ) \left (1 - \frac{2}{1 + \sqrt{3} i}\right ) \left (\sqrt{3} - i\right )}\right )} F\left (\operatorname{atan}{\left (x \left (\frac{\sqrt{3}}{2} - \frac{i}{2}\right ) \right )}\middle | \frac{3}{2} - \frac{\sqrt{3} i}{2}\right )}{\sqrt{\frac{x^{2} \left (\frac{1}{2} + \frac{\sqrt{3} i}{2}\right ) + 1}{x^{2} \left (\frac{1}{2} - \frac{\sqrt{3} i}{2}\right ) + 1}} \left (x^{2} \left (\frac{1}{2} - \frac{\sqrt{3} i}{2}\right ) + 1\right )} + \frac{\left (\frac{\sqrt{3}}{2} - \frac{i}{2}\right ) \sqrt{x^{4} + x^{2} + 1} \Pi \left (\frac{1}{2} - \frac{\sqrt{3} i}{2}; \operatorname{atan}{\left (x \left (\frac{\sqrt{3}}{2} - \frac{i}{2}\right ) \right )}\middle | \frac{3}{2} - \frac{\sqrt{3} i}{2}\right )}{\sqrt{\frac{x^{2} \left (\frac{1}{2} + \frac{\sqrt{3} i}{2}\right ) + 1}{x^{2} \left (\frac{1}{2} - \frac{\sqrt{3} i}{2}\right ) + 1}} \left (x^{2} \left (\frac{1}{2} - \frac{\sqrt{3} i}{2}\right ) + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_e(2*atan(x), 1/4)/(2*s
qrt(x**4 + x**2 + 1)) - sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*ellipti
c_f(2*atan(x), 1/4)/(2*sqrt(x**4 + x**2 + 1)) - 4*I*sqrt(x**4 + x**2 + 1)*im(1/(
(1 + sqrt(3)*I)*(1 - 2/(1 + sqrt(3)*I))*(sqrt(3) - I)))*elliptic_f(atan(x*(sqrt(
3)/2 - I/2)), 3/2 - sqrt(3)*I/2)/(sqrt((x**2*(1/2 + sqrt(3)*I/2) + 1)/(x**2*(1/2
- sqrt(3)*I/2) + 1))*(x**2*(1/2 - sqrt(3)*I/2) + 1)) + (sqrt(3)/2 - I/2)*sqrt(x
**4 + x**2 + 1)*elliptic_pi(1/2 - sqrt(3)*I/2, atan(x*(sqrt(3)/2 - I/2)), 3/2 -
sqrt(3)*I/2)/(sqrt((x**2*(1/2 + sqrt(3)*I/2) + 1)/(x**2*(1/2 - sqrt(3)*I/2) + 1)
)*(x**2*(1/2 - sqrt(3)*I/2) + 1))

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Mathematica [C]  time = 0.742727, size = 226, normalized size = 1.92 \[ \frac{-(-1)^{2/3} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-2 (-1)^{2/3} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \Pi \left (\sqrt [3]{-1};-i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+\sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left (F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )+\frac{x^5+x^3+x}{x^2+1}}{2 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.03, size = 397, normalized size = 3.4 \[{\frac{x}{2\,{x}^{2}+2}\sqrt{{x}^{4}+{x}^{2}+1}}-{\frac{1}{\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+2\,{\frac{\sqrt{1+1/2\,{x}^{2}-i/2{x}^{2}\sqrt{3}}\sqrt{1+1/2\,{x}^{2}+i/2{x}^{2}\sqrt{3}}{\it EllipticF} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{3}},1/2\,\sqrt{-2+2\,i\sqrt{3}} \right ) }{\sqrt{-2+2\,i\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1} \left ( i\sqrt{3}+1 \right ) }}-2\,{\frac{\sqrt{1+1/2\,{x}^{2}-i/2{x}^{2}\sqrt{3}}\sqrt{1+1/2\,{x}^{2}+i/2{x}^{2}\sqrt{3}}{\it EllipticE} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{3}},1/2\,\sqrt{-2+2\,i\sqrt{3}} \right ) }{\sqrt{-2+2\,i\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1} \left ( i\sqrt{3}+1 \right ) }}+{\frac{1}{\sqrt{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}}\sqrt{1+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}{x}^{2}\sqrt{3}}\sqrt{1+{\frac{{x}^{2}}{2}}+{\frac{i}{2}}{x}^{2}\sqrt{3}}{\it EllipticPi} \left ( \sqrt{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}x,- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3}}}{\sqrt{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*(x^4+x^2+1)^(1/2)/(x^2+1)-1/(-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/(-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/(-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))+1/(-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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((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. \[ \int \frac{1}{\sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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