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

Optimal. Leaf size=58 \[ \frac{1}{3} \sqrt{x^4+1} x+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+1}} \]

[Out]

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

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Rubi [A]  time = 0.0224954, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{3} \sqrt{x^4+1} x+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.42838, size = 49, normalized size = 0.84 \[ \frac{x \sqrt{x^{4} + 1}}{3} + \frac{\sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0329822, size = 48, normalized size = 0.83 \[ \frac{x^5-2 \sqrt [4]{-1} \sqrt{x^4+1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+x}{3 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x^4],x]

[Out]

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

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Maple [C]  time = 0.006, size = 72, normalized size = 1.2 \[{\frac{x}{3}\sqrt{{x}^{4}+1}}+{\frac{2\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{{\frac{3\,\sqrt{2}}{2}}+{\frac{3\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(x^4 + 1), x)

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Sympy [A]  time = 1.69543, size = 29, normalized size = 0.5 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+1)**(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^4 + 1), x)