3.929 \(\int \frac{1}{x^8 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{\sqrt{x^4+1}}{7 x^7}+\frac{5 \sqrt{x^4+1}}{21 x^3}+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{x^4+1}} \]

[Out]

-Sqrt[1 + x^4]/(7*x^7) + (5*Sqrt[1 + x^4])/(21*x^3) + (5*(1 + x^2)*Sqrt[(1 + x^4
)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(42*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0463073, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{x^4+1}}{7 x^7}+\frac{5 \sqrt{x^4+1}}{21 x^3}+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(7*x^7) + (5*Sqrt[1 + x^4])/(21*x^3) + (5*(1 + x^2)*Sqrt[(1 + x^4
)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(42*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 3.99967, size = 68, normalized size = 0.89 \[ \frac{5 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{42 \sqrt{x^{4} + 1}} + \frac{5 \sqrt{x^{4} + 1}}{21 x^{3}} - \frac{\sqrt{x^{4} + 1}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(x**4+1)**(1/2),x)

[Out]

5*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(42*sqrt(
x**4 + 1)) + 5*sqrt(x**4 + 1)/(21*x**3) - sqrt(x**4 + 1)/(7*x**7)

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Mathematica [C]  time = 0.0396248, size = 61, normalized size = 0.8 \[ \frac{5 x^8+2 x^4-5 \sqrt [4]{-1} \sqrt{x^4+1} x^7 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-3}{21 x^7 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*Sqrt[1 + x^4]),x]

[Out]

(-3 + 2*x^4 + 5*x^8 - 5*(-1)^(1/4)*x^7*Sqrt[1 + x^4]*EllipticF[I*ArcSinh[(-1)^(1
/4)*x], -1])/(21*x^7*Sqrt[1 + x^4])

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Maple [C]  time = 0.017, size = 86, normalized size = 1.1 \[ -{\frac{1}{7\,{x}^{7}}\sqrt{{x}^{4}+1}}+{\frac{5}{21\,{x}^{3}}\sqrt{{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{{\frac{21\,\sqrt{2}}{2}}+{\frac{21\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(x^4+1)^(1/2),x)

[Out]

-1/7*(x^4+1)^(1/2)/x^7+5/21*(x^4+1)^(1/2)/x^3+5/21/(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. \[ \int \frac{1}{\sqrt{x^{4} + 1} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^8),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^4 + 1)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{4} + 1} x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^8),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^4 + 1)*x^8), x)

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Sympy [A]  time = 3.59373, size = 36, normalized size = 0.47 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(x**4+1)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} + 1} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^8),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^4 + 1)*x^8), x)