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

Optimal. Leaf size=125 \[ -\frac{\sqrt{1-i x^2}}{2 (x+1)}-\frac{\sqrt{1+i x^2}}{2 (x+1)}-\frac{1}{4} (1-i)^{3/2} \tanh ^{-1}\left (\frac{1+i x}{\sqrt{1-i} \sqrt{1-i x^2}}\right )-\frac{1}{4} (1+i)^{3/2} \tanh ^{-1}\left (\frac{1-i x}{\sqrt{1+i} \sqrt{1+i x^2}}\right ) \]

[Out]

-Sqrt[1 - I*x^2]/(2*(1 + x)) - Sqrt[1 + I*x^2]/(2*(1 + x)) - ((1 - I)^(3/2)*ArcT
anh[(1 + I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])])/4 - ((1 + I)^(3/2)*ArcTanh[(1 - I*
x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/4

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Rubi [A]  time = 0.328194, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\sqrt{1-i x^2}}{2 (x+1)}-\frac{\sqrt{1+i x^2}}{2 (x+1)}-\frac{1}{4} (1-i)^{3/2} \tanh ^{-1}\left (\frac{1+i x}{\sqrt{1-i} \sqrt{1-i x^2}}\right )-\frac{1}{4} (1+i)^{3/2} \tanh ^{-1}\left (\frac{1-i x}{\sqrt{1+i} \sqrt{1+i x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - I*x^2]/(2*(1 + x)) - Sqrt[1 + I*x^2]/(2*(1 + x)) - ((1 - I)^(3/2)*ArcT
anh[(1 + I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])])/4 - ((1 + I)^(3/2)*ArcTanh[(1 - I*
x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/4

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Rubi in Sympy [A]  time = 21.5739, size = 202, normalized size = 1.62 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (- i x + 1\right )}{\sqrt{i x^{2} + 1} \left (\sqrt{1 + \sqrt{2}} + i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2 \left (- \sqrt{-1 + \sqrt{2}} + i \sqrt{1 + \sqrt{2}}\right )} - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (i x + 1\right )}{\sqrt{- i x^{2} + 1} \left (\sqrt{1 + \sqrt{2}} - i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2 \left (\sqrt{-1 + \sqrt{2}} + i \sqrt{1 + \sqrt{2}}\right )} - \frac{\left (\frac{1}{2} - \frac{i}{2}\right ) \left (\frac{1}{2} + \frac{i}{2}\right ) \sqrt{- i x^{2} + 1}}{x + 1} - \frac{\left (\frac{1}{2} - \frac{i}{2}\right ) \left (\frac{1}{2} + \frac{i}{2}\right ) \sqrt{i x^{2} + 1}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.154463, size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{(1+x)^2 \sqrt{1+x^4}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 1+x \right ) ^{2}}\sqrt{{x}^{2}+\sqrt{{x}^{4}+1}}{\frac{1}{\sqrt{{x}^{4}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)^2),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)^2),x, algorithm="giac")

[Out]

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