3.1525 \(\int \frac{x}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=45 \[ \frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{4 \sqrt{x^8+1}} \]

[Out]

((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcTan[x^2], 1/2])/(4*Sqrt[1
+ x^8])

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Rubi [A]  time = 0.0434147, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{4 \sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[1 + x^8],x]

[Out]

((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcTan[x^2], 1/2])/(4*Sqrt[1
+ x^8])

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Rubi in Sympy [A]  time = 2.49572, size = 39, normalized size = 0.87 \[ \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{8} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**8+1)**(1/2),x)

[Out]

sqrt((x**8 + 1)/(x**4 + 1)**2)*(x**4 + 1)*elliptic_f(2*atan(x**2), 1/2)/(4*sqrt(
x**8 + 1))

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Mathematica [C]  time = 0.0191487, size = 22, normalized size = 0.49 \[ \frac{1}{2} x^2 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[1 + x^8],x]

[Out]

(x^2*Hypergeometric2F1[1/4, 1/2, 5/4, -x^8])/2

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Maple [C]  time = 0.02, size = 17, normalized size = 0.4 \[{\frac{{x}^{2}}{2}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{5}{4}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^8+1)^(1/2),x)

[Out]

1/2*x^2*hypergeom([1/4,1/2],[5/4],-x^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

integrate(x/sqrt(x^8 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{x^{8} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

integral(x/sqrt(x^8 + 1), x)

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Sympy [A]  time = 1.64347, size = 29, normalized size = 0.64 \[ \frac{x^{2} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**8+1)**(1/2),x)

[Out]

x**2*gamma(1/4)*hyper((1/4, 1/2), (5/4,), x**8*exp_polar(I*pi))/(8*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

integrate(x/sqrt(x^8 + 1), x)