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

Optimal. Leaf size=60 \[ -\frac{\sqrt{x^4+1}}{3 x^3}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{6 \sqrt{x^4+1}} \]

[Out]

-Sqrt[1 + x^4]/(3*x^3) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcT
an[x], 1/2])/(6*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0295383, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{x^4+1}}{3 x^3}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{6 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(3*x^3) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcT
an[x], 1/2])/(6*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 2.76526, size = 53, normalized size = 0.88 \[ - \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{6 \sqrt{x^{4} + 1}} - \frac{\sqrt{x^{4} + 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0339854, size = 55, normalized size = 0.92 \[ \frac{-x^4+\sqrt [4]{-1} \sqrt{x^4+1} x^3 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-1}{3 x^3 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.013, size = 74, normalized size = 1.2 \[ -{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{4}+1}}-{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{3\,\sqrt{2}}{2}}+{\frac{3\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^4+1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^4 + 1)*x^4), x)

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Sympy [A]  time = 2.21277, size = 32, normalized size = 0.53 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^4), x)