\(\int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx\) [2655]

Optimal result

Integrand size = 46, antiderivative size = 236 \[ \int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\left (31736+1545 x+112192 x^2-2560 x^3+40320 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-1542+40688 x+1536 x^2+2240 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (2825+92032 x-2560 x^2+40320 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (39568+1536 x+2240 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{55440 \left (x+\sqrt {1+x^2}\right )^{3/2}}-\frac {1}{16} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \]

[Out]

1/55440*((40320*x^4-2560*x^3+112192*x^2+1545*x+31736)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(2240*x^3+1536*x^2+406
88*x-1542)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(x^2+1)^(1/2)*((40320*x^3-2560*x^2+92032*
x+2825)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(2240*x^2+1536*x+39568)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))
^(1/2))^(1/2)))/(x+(x^2+1)^(1/2))^(3/2)-1/16*arctanh((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2))

Rubi [F]

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

[In]

Int[Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]

[Out]

Defer[Int][Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]], x]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72 \[ \int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (31736+1545 x+112192 x^2-2560 x^3+40320 x^4+2 \left (-771+20344 x+768 x^2+1120 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (2825+92032 x-2560 x^2+40320 x^3+16 \left (2473+96 x+140 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{55440 \left (x+\sqrt {1+x^2}\right )^{3/2}}-\frac {1}{16} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \]

[In]

Integrate[Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]

[Out]

(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(31736 + 1545*x + 112192*x^2 - 2560*x^3 + 40320*x^4 + 2*(-771 + 20344*x + 7
68*x^2 + 1120*x^3)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(2825 + 92032*x - 2560*x^2 + 40320*x^3 + 16*(2473 +
96*x + 140*x^2)*Sqrt[x + Sqrt[1 + x^2]])))/(55440*(x + Sqrt[1 + x^2])^(3/2)) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt
[1 + x^2]]]]/16

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.50 \[ \int \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{55440} \, {\left (1120 \, x^{2} + 2 \, \sqrt {x^{2} + 1} {\left (560 \, x - 771\right )} - {\left (8400 \, x^{2} - 5 \, \sqrt {x^{2} + 1} {\left (5712 \, x + 565\right )} + 4105 \, x - 31736\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 3078 \, x + 39568\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {1}{32} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {1}{32} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \]

[In]

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

[Out]

1/55440*(1120*x^2 + 2*sqrt(x^2 + 1)*(560*x - 771) - (8400*x^2 - 5*sqrt(x^2 + 1)*(5712*x + 565) + 4105*x - 3173
6)*sqrt(x + sqrt(x^2 + 1)) + 3078*x + 39568)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1/32*log(sqrt(sqrt(x + sqrt(x
^2 + 1)) + 1) + 1) + 1/32*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1)

Sympy [F]

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

[In]

integrate((x**2+1)**(1/2)*(x+(x**2+1)**(1/2))**(1/2)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((x^2+1)^(1/2)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

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

[In]

integrate((x^2+1)^(1/2)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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