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

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

[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 = 0.0221527, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x}{2 \sqrt{x^4+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[(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 = 1.37587, 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(1/(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*at
an(x), 1/2)/(4*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.0318901, size = 37, normalized size = 0.64 \[ \frac{1}{2} \left (\frac{x}{\sqrt{x^4+1}}-\sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^4)^(-3/2),x]

[Out]

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

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Maple [C]  time = 0.008, 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(1/(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. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4 + 1)^(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((x^4 + 1)^(-3/2), x)

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Sympy [A]  time = 1.80607, size = 27, normalized size = 0.47 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4+1)**(3/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4 + 1)^(-3/2), x)