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

Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{x^4+1}+x^2}}\right )}{\sqrt{2}} \]

[Out]

ArcTanh[(Sqrt[2]*x)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]

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Rubi [A]  time = 0.088642, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{x^4+1}+x^2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x^2 + Sqrt[1 + x^4]]/Sqrt[1 + x^4],x]

[Out]

ArcTanh[(Sqrt[2]*x)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]

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Rubi in Sympy [A]  time = 3.71235, size = 29, normalized size = 0.94 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{x^{2} + \sqrt{x^{4} + 1}}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atanh(sqrt(2)*x/sqrt(x**2 + sqrt(x**4 + 1)))/2

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Mathematica [B]  time = 2.35662, size = 145, normalized size = 4.68 \[ -\frac{x \left (x^4+\sqrt{x^4+1} x^2+1\right ) \left (\log \left (1-\frac{\sqrt{x^2 \left (\sqrt{x^4+1}+x^2\right )}}{\sqrt{2} x^2}\right )-\log \left (\frac{\sqrt{x^2 \left (\sqrt{x^4+1}+x^2\right )}}{\sqrt{2} x^2}+1\right )\right )}{2 \sqrt{2} \sqrt{x^4+1} \sqrt{\sqrt{x^4+1}+x^2} \sqrt{x^2 \left (\sqrt{x^4+1}+x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(1 + x^4 + x^2*Sqrt[1 + x^4])*(Log[1 - Sqrt[x^2*(x^2 + Sqrt[1 + x^4])]/(Sqrt
[2]*x^2)] - Log[1 + Sqrt[x^2*(x^2 + Sqrt[1 + x^4])]/(Sqrt[2]*x^2)]))/(2*Sqrt[2]*
Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]]*Sqrt[x^2*(x^2 + Sqrt[1 + x^4])])

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int{1\sqrt{{x}^{2}+\sqrt{{x}^{4}+1}}{\frac{1}{\sqrt{{x}^{4}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)

[Out]

int((x^2+(x^4+1)^(1/2))^(1/2)/(x^4+1)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^2 + sqrt(x^4 + 1))/sqrt(x^4 + 1), x)

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Fricas [A]  time = 0.389951, size = 81, normalized size = 2.61 \[ \frac{1}{4} \, \sqrt{2} \log \left (4 \, x^{4} + 4 \, \sqrt{x^{4} + 1} x^{2} + 2 \,{\left (\sqrt{2} x^{3} + \sqrt{2} \sqrt{x^{4} + 1} x\right )} \sqrt{x^{2} + \sqrt{x^{4} + 1}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4
+ 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1)

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Sympy [A]  time = 2.21216, size = 15, normalized size = 0.48 \[ \frac{{G_{3, 3}^{2, 2}\left (\begin{matrix} 1, 1 & \frac{1}{2} \\\frac{1}{4}, \frac{3}{4} & 0 \end{matrix} \middle |{x^{4}} \right )}}{4 \sqrt{\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

meijerg(((1, 1), (1/2,)), ((1/4, 3/4), (0,)), x**4)/(4*sqrt(pi))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^2 + sqrt(x^4 + 1))/sqrt(x^4 + 1), x)