\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\) [576]

Optimal result

Integrand size = 27, antiderivative size = 44 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {\log \left (x^2+\sqrt {1+x^4}+\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[Out]

1/2*ln(x^2+(x^4+1)^(1/2)+2^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2157, 212} \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}} \]

[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 212

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

Rule 2157

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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {\log \left (x^2+\sqrt {1+x^4}+\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.98 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\frac {{G_{3, 3}^{2, 2}\left (\begin {matrix} 1, 1 & \frac {1}{2} \\\frac {1}{4}, \frac {3}{4} & 0 \end {matrix} \middle | {x^{4}} \right )}}{4 \sqrt {\pi }} \]

[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))

Maxima [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}} \,d x \]

[In]

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

[Out]

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