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

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

[Out]

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

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Rubi [A]  time = 0.0544915, 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, Rules used = {2132, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2+\sqrt{x^4+1}}}\right )}{\sqrt{2}} \]

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]

Rule 2132

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

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

Mathematica [A]  time = 0.0083347, size = 31, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2+\sqrt{x^4+1}}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[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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Maple [F]  time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{\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)/(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. \begin{align*} \int \frac{\sqrt{x^{2} + \sqrt{x^{4} + 1}}}{\sqrt{x^{4} + 1}}\,{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)/(x^4+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 4.026, size = 162, normalized size = 5.23 \begin{align*} \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 ) \end{align*}

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, 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 = 1.28076, size = 15, normalized size = 0.48 \begin{align*} \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 }} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + \sqrt{x^{4} + 1}}}{\sqrt{x^{4} + 1}}\,{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)/(x^4+1)^(1/2),x, algorithm="giac")

[Out]

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