\(\int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx\) [2654]

Optimal result

Integrand size = 46, antiderivative size = 236 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\left (-24993-2680 x-21570 x^2-4096 x^3+30720 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (1712+1814 x+4096 x^2+3072 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-632-36930 x-4096 x^2+30720 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (278+4096 x+3072 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{26880 \left (x+\sqrt {1+x^2}\right )^{5/2}}-\frac {263}{256} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \]

[Out]

1/26880*((30720*x^4-4096*x^3-21570*x^2-2680*x-24993)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(3072*x^3+4096*x^2+1814
*x+1712)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(x^2+1)^(1/2)*((30720*x^3-4096*x^2-36930*x-
632)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(3072*x^2+4096*x+278)*(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))^(5/2)-263/256*arctanh((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2))

Rubi [F]

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

[In]

Int[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[x + Sqrt[1 + x^2]],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-24993-2680 x-21570 x^2-4096 x^3+30720 x^4+2 \left (856+907 x+2048 x^2+1536 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-632-36930 x-4096 x^2+30720 x^3+\left (278+4096 x+3072 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (x+\sqrt {1+x^2}\right )^{5/2}}-27615 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{26880} \]

[In]

Integrate[(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[x + Sqrt[1 + x^2]],x]

[Out]

((Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-24993 - 2680*x - 21570*x^2 - 4096*x^3 + 30720*x^4 + 2*(856 + 907*x + 204
8*x^2 + 1536*x^3)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-632 - 36930*x - 4096*x^2 + 30720*x^3 + (278 + 4096
*x + 3072*x^2)*Sqrt[x + Sqrt[1 + x^2]])))/(x + Sqrt[1 + x^2])^(5/2) - 27615*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 +
x^2]]]])/26880

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{26880} \, {\left (672 \, x^{2} - 2 \, \sqrt {x^{2} + 1} {\left (336 \, x + 139\right )} - {\left (10752 \, x^{3} + 784 \, x^{2} - {\left (10752 \, x^{2} + 784 \, x + 24993\right )} \sqrt {x^{2} + 1} + 38049 \, x - 632\right )} \sqrt {x + \sqrt {x^{2} + 1}} - 1258 \, x - 1712\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {263}{512} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {263}{512} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \]

[In]

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

[Out]

-1/26880*(672*x^2 - 2*sqrt(x^2 + 1)*(336*x + 139) - (10752*x^3 + 784*x^2 - (10752*x^2 + 784*x + 24993)*sqrt(x^
2 + 1) + 38049*x - 632)*sqrt(x + sqrt(x^2 + 1)) - 1258*x - 1712)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 263/512*l
og(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) + 263/512*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[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)