\(\int \sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx\) [2681]

Optimal result

Integrand size = 31, antiderivative size = 242 \[ \int \sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\left (78032-1254 x+193024 x^2-3072 x^3+35840 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (184+345 x+2048 x^2+2560 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (282+175104 x-3072 x^2+35840 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-935+2048 x+2560 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{80640 x \sqrt {1+x^2}+40320 \left (1+2 x^2\right )}-\frac {251}{128} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \]

[Out]

((35840*x^4-3072*x^3+193024*x^2-1254*x+78032)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(2560*x^3+2048*x^2+345*x+184)*
(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(x^2+1)^(1/2)*((35840*x^3-3072*x^2+175104*x+282)*(1+
(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(2560*x^2+2048*x-935)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)
))/(80640*x*(x^2+1)^(1/2)+80640*x^2+40320)-251/128*arctanh((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2))

Rubi [F]

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

[In]

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

[Out]

Defer[Int][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 {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.73 \[ \int \sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (2 \left (39016-627 x+96512 x^2-1536 x^3+17920 x^4\right )+\left (184+345 x+2048 x^2+2560 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (282+175104 x-3072 x^2+35840 x^3+\left (-935+2048 x+2560 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{1+2 x^2+2 x \sqrt {1+x^2}}-79065 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{40320} \]

[In]

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

[Out]

((Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(2*(39016 - 627*x + 96512*x^2 - 1536*x^3 + 17920*x^4) + (184 + 345*x + 204
8*x^2 + 2560*x^3)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(282 + 175104*x - 3072*x^2 + 35840*x^3 + (-935 + 204
8*x + 2560*x^2)*Sqrt[x + Sqrt[1 + x^2]])))/(1 + 2*x^2 + 2*x*Sqrt[1 + x^2]) - 79065*ArcTanh[Sqrt[1 + Sqrt[x + S
qrt[1 + x^2]]]])/40320

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.49 \[ \int \sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{40320} \, {\left (1120 \, x^{2} - 2 \, \sqrt {x^{2} + 1} {\left (9520 \, x + 141\right )} + {\left (1680 \, x^{2} - 5 \, \sqrt {x^{2} + 1} {\left (336 \, x - 187\right )} - 2215 \, x - 184\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 1818 \, x - 78032\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {251}{256} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {251}{256} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \]

[In]

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

[Out]

-1/40320*(1120*x^2 - 2*sqrt(x^2 + 1)*(9520*x + 141) + (1680*x^2 - 5*sqrt(x^2 + 1)*(336*x - 187) - 2215*x - 184
)*sqrt(x + sqrt(x^2 + 1)) + 1818*x - 78032)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 251/256*log(sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1) + 1) + 251/256*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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