3.934 \(\int \frac{1}{x^6 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=140 \[ \frac{3 \sqrt{x^4+1}}{5 x}-\frac{\sqrt{x^4+1}}{5 x^5}-\frac{3 \sqrt{x^4+1} x}{5 \left (x^2+1\right )}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+1}} \]

[Out]

-Sqrt[1 + x^4]/(5*x^5) + (3*Sqrt[1 + x^4])/(5*x) - (3*x*Sqrt[1 + x^4])/(5*(1 + x
^2)) + (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(5*
Sqrt[1 + x^4]) - (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x],
1/2])/(10*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0809755, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 \sqrt{x^4+1}}{5 x}-\frac{\sqrt{x^4+1}}{5 x^5}-\frac{3 \sqrt{x^4+1} x}{5 \left (x^2+1\right )}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(5*x^5) + (3*Sqrt[1 + x^4])/(5*x) - (3*x*Sqrt[1 + x^4])/(5*(1 + x
^2)) + (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(5*
Sqrt[1 + x^4]) - (3*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x],
1/2])/(10*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 8.14026, size = 126, normalized size = 0.9 \[ - \frac{3 x \sqrt{x^{4} + 1}}{5 \left (x^{2} + 1\right )} + \frac{3 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{x^{4} + 1}} - \frac{3 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{10 \sqrt{x^{4} + 1}} + \frac{3 \sqrt{x^{4} + 1}}{5 x} - \frac{\sqrt{x^{4} + 1}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x*sqrt(x**4 + 1)/(5*(x**2 + 1)) + 3*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)
*elliptic_e(2*atan(x), 1/2)/(5*sqrt(x**4 + 1)) - 3*sqrt((x**4 + 1)/(x**2 + 1)**2
)*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(10*sqrt(x**4 + 1)) + 3*sqrt(x**4 + 1)/(
5*x) - sqrt(x**4 + 1)/(5*x**5)

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Mathematica [C]  time = 0.0530177, size = 94, normalized size = 0.67 \[ \frac{3 x^8+2 x^4-3 (-1)^{3/4} \sqrt{x^4+1} x^5 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+3 (-1)^{3/4} \sqrt{x^4+1} x^5 E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-1}{5 x^5 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.015, size = 107, normalized size = 0.8 \[ -{\frac{1}{5\,{x}^{5}}\sqrt{{x}^{4}+1}}+{\frac{3}{5\,x}\sqrt{{x}^{4}+1}}-{\frac{{\frac{3\,i}{5}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^4+1)^(1/2),x)

[Out]

-1/5*(x^4+1)^(1/2)/x^5+3/5*(x^4+1)^(1/2)/x-3/5*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. \[ \int \frac{1}{\sqrt{x^{4} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^4 + 1)*x^6), x)

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Sympy [A]  time = 2.71082, size = 36, normalized size = 0.26 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-5/4)*hyper((-5/4, 1/2), (-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. \[ \int \frac{1}{\sqrt{x^{4} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^6), x)