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 = 0.16247, 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, Rules used = {2133, 725, 206} \[ -\frac{1}{2} \sqrt{1-i} \tanh ^{-1}\left (\frac{1+i x}{\sqrt{1-i} \sqrt{1-i x^2}}\right )-\frac{1}{2} \sqrt{1+i} \tanh ^{-1}\left (\frac{1-i x}{\sqrt{1+i} \sqrt{1+i x^2}}\right ) \]

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

Rule 2133

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_S
ymbol] :> Dist[(1 - I)/2, Int[(c + d*x)^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Dist[(1 + I)/2, Int[(c + d*x)^m/Sq
rt[Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && GtQ[a, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{x^2+\sqrt{1+x^4}}}{(1+x) \sqrt{1+x^4}} \, dx &=\left (\frac{1}{2}-\frac{i}{2}\right ) \int \frac{1}{(1+x) \sqrt{1-i x^2}} \, dx+\left (\frac{1}{2}+\frac{i}{2}\right ) \int \frac{1}{(1+x) \sqrt{1+i x^2}} \, dx\\ &=\left (-\frac{1}{2}-\frac{i}{2}\right ) \operatorname{Subst}\left (\int \frac{1}{(1+i)-x^2} \, dx,x,\frac{1-i x}{\sqrt{1+i x^2}}\right )+\left (-\frac{1}{2}+\frac{i}{2}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-i)-x^2} \, dx,x,\frac{1+i x}{\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 )-\frac{1}{2} \sqrt{1+i} \tanh ^{-1}\left (\frac{1-i x}{\sqrt{1+i} \sqrt{1+i x^2}}\right )\\ \end{align*}

Mathematica [F]  time = 0.175985, 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 = 0.022, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{1+x}\sqrt{{x}^{2}+\sqrt{{x}^{4}+1}}{\frac{1}{\sqrt{{x}^{4}+1}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{\sqrt{x^{2} + \sqrt{x^{4} + 1}}}{\sqrt{x^{4} + 1}{\left (x + 1\right )}}\,{d x} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 38.9638, size = 998, normalized size = 12.32 \begin{align*} \frac{1}{2} \, \sqrt{2 \, \sqrt{2} - 2} \arctan \left (\frac{{\left (2 \, x^{2} - \sqrt{2}{\left (x^{3} - x^{2} + x + 1\right )} + \sqrt{x^{4} + 1}{\left (\sqrt{2}{\left (x - 1\right )} - 2\right )} - 2 \, x\right )} \sqrt{x^{2} + \sqrt{x^{4} + 1}} \sqrt{2 \, \sqrt{2} - 2} +{\left (x^{2} + \sqrt{2} \sqrt{x^{4} + 1} + 1\right )} \sqrt{2 \, \sqrt{2} + 2} \sqrt{2 \, \sqrt{2} - 2}}{2 \,{\left (x^{2} - 2 \, x + 1\right )}}\right ) - \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 2} \log \left (-\frac{{\left (2 \, x^{3} - \sqrt{2}{\left (x^{3} - x^{2} - x - 1\right )} + \sqrt{x^{4} + 1}{\left (\sqrt{2}{\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt{x^{2} + \sqrt{x^{4} + 1}} +{\left (x^{2} - \sqrt{2}{\left (x^{2} + 1\right )} + \sqrt{x^{4} + 1}{\left (\sqrt{2} - 2\right )} + 1\right )} \sqrt{2 \, \sqrt{2} + 2}}{x^{2} + 2 \, x + 1}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 2} \log \left (-\frac{{\left (2 \, x^{3} - \sqrt{2}{\left (x^{3} - x^{2} - x - 1\right )} + \sqrt{x^{4} + 1}{\left (\sqrt{2}{\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt{x^{2} + \sqrt{x^{4} + 1}} -{\left (x^{2} - \sqrt{2}{\left (x^{2} + 1\right )} + \sqrt{x^{4} + 1}{\left (\sqrt{2} - 2\right )} + 1\right )} \sqrt{2 \, \sqrt{2} + 2}}{x^{2} + 2 \, x + 1}\right ) \end{align*}

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, algorithm="fricas")

[Out]

1/2*sqrt(2*sqrt(2) - 2)*arctan(1/2*((2*x^2 - sqrt(2)*(x^3 - x^2 + x + 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2)
- 2*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(2*sqrt(2) - 2) + (x^2 + sqrt(2)*sqrt(x^4 + 1) + 1)*sqrt(2*sqrt(2) + 2)*
sqrt(2*sqrt(2) - 2))/(x^2 - 2*x + 1)) - 1/8*sqrt(2*sqrt(2) + 2)*log(-((2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + s
qrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) + (x^2 - sqrt(2)*(x^2 + 1) + sqrt(x^4 + 1)
*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2 + 2*x + 1)) + 1/8*sqrt(2*sqrt(2) + 2)*log(-((2*x^3 - sqrt(2)*(x^
3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (x^2 - sqrt(2)*(x^2
+ 1) + sqrt(x^4 + 1)*(sqrt(2) - 2) + 1)*sqrt(2*sqrt(2) + 2))/(x^2 + 2*x + 1))

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

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

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, algorithm="giac")

[Out]

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