3.950 \(\int \frac{1}{x^8 \left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{9 \sqrt{x^4+1}}{14 x^7}+\frac{1}{2 x^7 \sqrt{x^4+1}}+\frac{15 \sqrt{x^4+1}}{14 x^3}+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]

[Out]

1/(2*x^7*Sqrt[1 + x^4]) - (9*Sqrt[1 + x^4])/(14*x^7) + (15*Sqrt[1 + x^4])/(14*x^
3) + (15*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(28*
Sqrt[1 + x^4])

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Rubi [A]  time = 0.0611315, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{9 \sqrt{x^4+1}}{14 x^7}+\frac{1}{2 x^7 \sqrt{x^4+1}}+\frac{15 \sqrt{x^4+1}}{14 x^3}+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(1 + x^4)^(3/2)),x]

[Out]

1/(2*x^7*Sqrt[1 + x^4]) - (9*Sqrt[1 + x^4])/(14*x^7) + (15*Sqrt[1 + x^4])/(14*x^
3) + (15*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(28*
Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 5.49764, size = 85, normalized size = 0.92 \[ \frac{15 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{28 \sqrt{x^{4} + 1}} + \frac{15 \sqrt{x^{4} + 1}}{14 x^{3}} - \frac{9 \sqrt{x^{4} + 1}}{14 x^{7}} + \frac{1}{2 x^{7} \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(x**4+1)**(3/2),x)

[Out]

15*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(28*sqrt
(x**4 + 1)) + 15*sqrt(x**4 + 1)/(14*x**3) - 9*sqrt(x**4 + 1)/(14*x**7) + 1/(2*x*
*7*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.042455, size = 61, normalized size = 0.66 \[ \frac{15 x^8+6 x^4-15 \sqrt [4]{-1} \sqrt{x^4+1} x^7 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-2}{14 x^7 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(1 + x^4)^(3/2)),x]

[Out]

(-2 + 6*x^4 + 15*x^8 - 15*(-1)^(1/4)*x^7*Sqrt[1 + x^4]*EllipticF[I*ArcSinh[(-1)^
(1/4)*x], -1])/(14*x^7*Sqrt[1 + x^4])

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Maple [C]  time = 0.02, size = 96, normalized size = 1. \[{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{7\,{x}^{7}}\sqrt{{x}^{4}+1}}+{\frac{4}{7\,{x}^{3}}\sqrt{{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{7\,\sqrt{2}+7\,i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(x^4+1)^(3/2),x)

[Out]

1/2*x/(x^4+1)^(1/2)-1/7*(x^4+1)^(1/2)/x^7+4/7*(x^4+1)^(1/2)/x^3+15/14/(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. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^8),x, algorithm="maxima")

[Out]

integrate(1/((x^4 + 1)^(3/2)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{12} + x^{8}\right )} \sqrt{x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^8),x, algorithm="fricas")

[Out]

integral(1/((x^12 + x^8)*sqrt(x^4 + 1)), x)

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Sympy [A]  time = 5.82385, size = 36, normalized size = 0.39 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(x**4+1)**(3/2),x)

[Out]

gamma(-7/4)*hyper((-7/4, 3/2), (-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. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^8),x, algorithm="giac")

[Out]

integrate(1/((x^4 + 1)^(3/2)*x^8), x)