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

Optimal. Leaf size=81 \[ -\frac{1}{2} \sqrt{1-i} \tanh ^{-1}\left (\frac{1+i x}{\sqrt{1-i} \sqrt{1-i x^2}}\right )-\frac{1}{2} \sqrt{1+i} \tanh ^{-1}\left (\frac{1-i x}{\sqrt{1+i} \sqrt{1+i x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.278117, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ -\frac{1}{2} \sqrt{1-i} \tanh ^{-1}\left (\frac{1+i x}{\sqrt{1-i} \sqrt{1-i x^2}}\right )-\frac{1}{2} \sqrt{1+i} \tanh ^{-1}\left (\frac{1-i x}{\sqrt{1+i} \sqrt{1+i x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.1132, size = 155, normalized size = 1.91 \[ - \frac{\sqrt{2} \left (1 + i\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (- i x + 1\right )}{\sqrt{i x^{2} + 1} \left (\sqrt{1 + \sqrt{2}} + i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2 \left (\sqrt{1 + \sqrt{2}} + i \sqrt{-1 + \sqrt{2}}\right )} + \frac{\sqrt{2} \left (1 - i\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (i x + 1\right )}{\sqrt{- i x^{2} + 1} \left (\sqrt{1 + \sqrt{2}} - i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2 \left (- \sqrt{1 + \sqrt{2}} + i \sqrt{-1 + \sqrt{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(1 + I)*atanh(sqrt(2)*(-I*x + 1)/(sqrt(I*x**2 + 1)*(sqrt(1 + sqrt(2)) +
I*sqrt(-1 + sqrt(2)))))/(2*(sqrt(1 + sqrt(2)) + I*sqrt(-1 + sqrt(2)))) + sqrt(2
)*(1 - I)*atanh(sqrt(2)*(I*x + 1)/(sqrt(-I*x**2 + 1)*(sqrt(1 + sqrt(2)) - I*sqrt
(-1 + sqrt(2)))))/(2*(-sqrt(1 + sqrt(2)) + I*sqrt(-1 + sqrt(2))))

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Mathematica [A]  time = 0.118799, size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{(1+x) \sqrt{1+x^4}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{1}{1+x}\sqrt{{x}^{2}+\sqrt{{x}^{4}+1}}{\frac{1}{\sqrt{{x}^{4}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x**2 + sqrt(x**4 + 1))/((x + 1)*sqrt(x**4 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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