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

Optimal. Leaf size=76 \[ -\frac{5 \sqrt{x^4+1}}{6 x^3}+\frac{1}{2 x^3 \sqrt{x^4+1}}-\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{12 \sqrt{x^4+1}} \]

[Out]

1/(2*x^3*Sqrt[1 + x^4]) - (5*Sqrt[1 + x^4])/(6*x^3) - (5*(1 + x^2)*Sqrt[(1 + x^4
)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(12*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0448991, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{5 \sqrt{x^4+1}}{6 x^3}+\frac{1}{2 x^3 \sqrt{x^4+1}}-\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{12 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 + x^4)^(3/2)),x]

[Out]

1/(2*x^3*Sqrt[1 + x^4]) - (5*Sqrt[1 + x^4])/(6*x^3) - (5*(1 + x^2)*Sqrt[(1 + x^4
)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(12*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 4.18499, size = 70, normalized size = 0.92 \[ - \frac{5 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{x^{4} + 1}} - \frac{5 \sqrt{x^{4} + 1}}{6 x^{3}} + \frac{1}{2 x^{3} \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(12*sqrt
(x**4 + 1)) - 5*sqrt(x**4 + 1)/(6*x**3) + 1/(2*x**3*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.0664525, size = 46, normalized size = 0.61 \[ \frac{1}{6} \left (\frac{-5 x^4-2}{x^3 \sqrt{x^4+1}}+5 \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2 - 5*x^4)/(x^3*Sqrt[1 + x^4]) + 5*(-1)^(1/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*
x], -1])/6

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Maple [C]  time = 0.017, size = 84, normalized size = 1.1 \[ -{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{4}+1}}-{\frac{5\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{3\,\sqrt{2}+3\,i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^4+1)^(3/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^4 + 1)^(3/2)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.83842, size = 32, normalized size = 0.42 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^4 + 1)^(3/2)*x^4), x)