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

Optimal. Leaf size=43 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]

[Out]

((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 = 0.012928, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + x^4],x]

[Out]

((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 = 1.15466, size = 37, normalized size = 0.86 \[ \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [C]  time = 0.0190761, size = 21, normalized size = 0.49 \[ -\sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 + x^4],x]

[Out]

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

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Maple [C]  time = 0.006, size = 60, normalized size = 1.4 \[{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(x^4 + 1), x)

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Sympy [A]  time = 1.60933, size = 27, normalized size = 0.63 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4+1)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(x^4 + 1), x)