3.933 \(\int \frac{1}{x^2 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{\sqrt{x^4+1}}{x}+\frac{\sqrt{x^4+1} x}{x^2+1}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+1}} \]

[Out]

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

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Rubi [A]  time = 0.0596317, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{\sqrt{x^4+1}}{x}+\frac{\sqrt{x^4+1} x}{x^2+1}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[1 + x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.71618, size = 100, normalized size = 0.85 \[ \frac{x \sqrt{x^{4} + 1}}{x^{2} + 1} - \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} + 1}} + \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{4} + 1}} - \frac{\sqrt{x^{4} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**4 + 1)/(x**2 + 1) - sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic
_e(2*atan(x), 1/2)/sqrt(x**4 + 1) + sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*el
liptic_f(2*atan(x), 1/2)/(2*sqrt(x**4 + 1)) - sqrt(x**4 + 1)/x

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Mathematica [C]  time = 0.064619, size = 70, normalized size = 0.6 \[ -\frac{1}{\sqrt{x^4+1} x}-\frac{x^3}{\sqrt{x^4+1}}+(-1)^{3/4} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-(-1)^{3/4} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.012, size = 95, normalized size = 0.8 \[ -{\frac{1}{x}\sqrt{{x}^{4}+1}}+{\frac{i \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^4+1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{4} + 1} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^4 + 1)*x^2), x)

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Sympy [A]  time = 1.9442, size = 31, normalized size = 0.26 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^4 + 1)*x^2), x)