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

Optimal. Leaf size=103 \[ \frac{\sqrt{x^4+1} x}{x^2+1}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+1}} \]

[Out]

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

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Rubi [A]  time = 0.048616, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{x^4+1} x}{x^2+1}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[1 + x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.588, size = 90, normalized size = 0.87 \[ \frac{x \sqrt{x^{4} + 1}}{x^{2} + 1} - \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} + 1}} + \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**4 + 1)/(x**2 + 1) - sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic
_e(2*atan(x), 1/2)/sqrt(x**4 + 1) + sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*el
liptic_f(2*atan(x), 1/2)/(2*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.0309104, size = 37, normalized size = 0.36 \[ (-1)^{3/4} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[1 + x^4],x]

[Out]

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

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Maple [C]  time = 0.008, size = 82, normalized size = 0.8 \[{\frac{i \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^4+1)^(1/2),x)

[Out]

I/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*(Ell
ipticF(x*(1/2*2^(1/2)+1/2*I*2^(1/2)),I)-EllipticE(x*(1/2*2^(1/2)+1/2*I*2^(1/2)),
I))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/sqrt(x^4 + 1), x)

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Sympy [A]  time = 1.70592, size = 29, normalized size = 0.28 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**4*exp_polar(I*pi))/(4*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/sqrt(x^4 + 1), x)