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

Optimal. Leaf size=124 \[ \frac{x^3}{2 \sqrt{x^4+1}}-\frac{\sqrt{x^4+1} x}{2 \left (x^2+1\right )}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]

[Out]

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

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Rubi [A]  time = 0.0726176, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{x^3}{2 \sqrt{x^4+1}}-\frac{\sqrt{x^4+1} x}{2 \left (x^2+1\right )}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 + x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.19606, size = 107, normalized size = 0.86 \[ \frac{x^{3}}{2 \sqrt{x^{4} + 1}} - \frac{x \sqrt{x^{4} + 1}}{2 \left (x^{2} + 1\right )} + \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{4} + 1}} - \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.063085, size = 59, normalized size = 0.48 \[ \frac{1}{2} \left (\frac{x^3}{\sqrt{x^4+1}}-(-1)^{3/4} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+(-1)^{3/4} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.01, size = 95, normalized size = 0.8 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{{\frac{i}{2}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^4+1)^(3/2),x)

[Out]

1/2*x^3/(x^4+1)^(1/2)-1/2*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)*(EllipticF(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. \[ \int \frac{x^{2}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(x^4 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(x^4 + 1)^(3/2), x)

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Sympy [A]  time = 1.87581, size = 29, normalized size = 0.23 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*gamma(3/4)*hyper((3/4, 3/2), (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. \[ \int \frac{x^{2}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(x^4 + 1)^(3/2), x)