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

Optimal. Leaf size=58 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}-\frac{x}{2 \sqrt{x^4+1}} \]

[Out]

-x/(2*Sqrt[1 + x^4]) + ((1 + x^2)*Sqrt[(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 = 0.0300925, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}-\frac{x}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(1 + x^4)^(3/2),x]

[Out]

-x/(2*Sqrt[1 + x^4]) + ((1 + x^2)*Sqrt[(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 = 2.82101, size = 49, normalized size = 0.84 \[ - \frac{x}{2 \sqrt{x^{4} + 1}} + \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.037637, size = 38, normalized size = 0.66 \[ -\frac{x}{2 \sqrt{x^4+1}}-\frac{1}{2} \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.011, size = 72, normalized size = 1.2 \[ -{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{\sqrt{2}+i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^4+1)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(x^4 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(x^4 + 1)^(3/2), x)

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Sympy [A]  time = 1.92442, size = 29, normalized size = 0.5 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), x**4*exp_polar(I*pi))/(4*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(x^4 + 1)^(3/2), x)