3.956 \(\int \frac{1}{x^6 \left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=156 \[ \frac{21 \sqrt{x^4+1}}{10 x}-\frac{7 \sqrt{x^4+1}}{10 x^5}+\frac{1}{2 \sqrt{x^4+1} x^5}-\frac{21 \sqrt{x^4+1} x}{10 \left (x^2+1\right )}-\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{20 \sqrt{x^4+1}}+\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}} \]

[Out]

1/(2*x^5*Sqrt[1 + x^4]) - (7*Sqrt[1 + x^4])/(10*x^5) + (21*Sqrt[1 + x^4])/(10*x)
- (21*x*Sqrt[1 + x^4])/(10*(1 + x^2)) + (21*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^
2]*EllipticE[2*ArcTan[x], 1/2])/(10*Sqrt[1 + x^4]) - (21*(1 + x^2)*Sqrt[(1 + x^4
)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(20*Sqrt[1 + x^4])

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Rubi [A]  time = 0.104759, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{21 \sqrt{x^4+1}}{10 x}-\frac{7 \sqrt{x^4+1}}{10 x^5}+\frac{1}{2 \sqrt{x^4+1} x^5}-\frac{21 \sqrt{x^4+1} x}{10 \left (x^2+1\right )}-\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{20 \sqrt{x^4+1}}+\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 + x^4)^(3/2)),x]

[Out]

1/(2*x^5*Sqrt[1 + x^4]) - (7*Sqrt[1 + x^4])/(10*x^5) + (21*Sqrt[1 + x^4])/(10*x)
- (21*x*Sqrt[1 + x^4])/(10*(1 + x^2)) + (21*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^
2]*EllipticE[2*ArcTan[x], 1/2])/(10*Sqrt[1 + x^4]) - (21*(1 + x^2)*Sqrt[(1 + x^4
)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(20*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 9.79945, size = 143, normalized size = 0.92 \[ - \frac{21 x \sqrt{x^{4} + 1}}{10 \left (x^{2} + 1\right )} + \frac{21 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{10 \sqrt{x^{4} + 1}} - \frac{21 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{20 \sqrt{x^{4} + 1}} + \frac{21 \sqrt{x^{4} + 1}}{10 x} - \frac{7 \sqrt{x^{4} + 1}}{10 x^{5}} + \frac{1}{2 x^{5} \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**4+1)**(3/2),x)

[Out]

-21*x*sqrt(x**4 + 1)/(10*(x**2 + 1)) + 21*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 +
1)*elliptic_e(2*atan(x), 1/2)/(10*sqrt(x**4 + 1)) - 21*sqrt((x**4 + 1)/(x**2 +
1)**2)*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(20*sqrt(x**4 + 1)) + 21*sqrt(x**4
+ 1)/(10*x) - 7*sqrt(x**4 + 1)/(10*x**5) + 1/(2*x**5*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.0586494, size = 94, normalized size = 0.6 \[ \frac{21 x^8+14 x^4-21 (-1)^{3/4} \sqrt{x^4+1} x^5 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+21 (-1)^{3/4} \sqrt{x^4+1} x^5 E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-2}{10 x^5 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 + x^4)^(3/2)),x]

[Out]

(-2 + 14*x^4 + 21*x^8 + 21*(-1)^(3/4)*x^5*Sqrt[1 + x^4]*EllipticE[I*ArcSinh[(-1)
^(1/4)*x], -1] - 21*(-1)^(3/4)*x^5*Sqrt[1 + x^4]*EllipticF[I*ArcSinh[(-1)^(1/4)*
x], -1])/(10*x^5*Sqrt[1 + x^4])

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Maple [C]  time = 0.02, size = 119, normalized size = 0.8 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{5\,{x}^{5}}\sqrt{{x}^{4}+1}}+{\frac{8}{5\,x}\sqrt{{x}^{4}+1}}-{\frac{{\frac{21\,i}{10}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^4+1)^(3/2),x)

[Out]

1/2*x^3/(x^4+1)^(1/2)-1/5*(x^4+1)^(1/2)/x^5+8/5*(x^4+1)^(1/2)/x-21/10*I/(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)-EllipticE(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. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((x^4 + 1)^(3/2)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{10} + x^{6}\right )} \sqrt{x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^6),x, algorithm="fricas")

[Out]

integral(1/((x^10 + x^6)*sqrt(x^4 + 1)), x)

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Sympy [A]  time = 4.03422, size = 36, normalized size = 0.23 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{2} \\ - \frac{1}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**4+1)**(3/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 1)^(3/2)*x^6),x, algorithm="giac")

[Out]

integrate(1/((x^4 + 1)^(3/2)*x^6), x)