\(\int (1+x^2)^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx\) [2894]

Optimal result

Integrand size = 46, antiderivative size = 316 \[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\left (15903121112+5227043711 x+220397520304 x^2-1415707308 x^3+407581982720 x^4-11794907136 x^5+248171986944 x^6-2099249152 x^7+66913566720 x^8\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-1176816782+66830366096 x+1984342244 x^2+96561463296 x^3+6568280064 x^4+10550149120 x^5+1130364928 x^6+1968046080 x^7\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (2167822549+88760534448 x+3694527828 x^2+308588576768 x^3-10745282560 x^4+214715203584 x^5-2099249152 x^6+66913566720 x^7\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (21890925968-875910940 x+92024406016 x^2+6003097600 x^3+9566126080 x^4+1130364928 x^5+1968046080 x^6\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{39729930240 \left (x+\sqrt {1+x^2}\right )^{7/2}}-\frac {545 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{8192} \]

[Out]

1/39729930240*((66913566720*x^8-2099249152*x^7+248171986944*x^6-11794907136*x^5+407581982720*x^4-1415707308*x^
3+220397520304*x^2+5227043711*x+15903121112)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(1968046080*x^7+1130364928*x^6+
10550149120*x^5+6568280064*x^4+96561463296*x^3+1984342244*x^2+66830366096*x-1176816782)*(x+(x^2+1)^(1/2))^(1/2
)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(x^2+1)^(1/2)*((66913566720*x^7-2099249152*x^6+214715203584*x^5-1074528256
0*x^4+308588576768*x^3+3694527828*x^2+88760534448*x+2167822549)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)+(1968046080*
x^6+1130364928*x^5+9566126080*x^4+6003097600*x^3+92024406016*x^2-875910940*x+21890925968)*(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))^(7/2)-545/8192*arctanh((1+(x+(x^2+1)^(1/2))^(1/2))^(
1/2))

Rubi [F]

\[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \]

[In]

Int[(1 + x^2)^(3/2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]

[Out]

Defer[Int][(1 + x^2)^(3/2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.79 \[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (15903121112+5227043711 x+220397520304 x^2-1415707308 x^3+407581982720 x^4-11794907136 x^5+248171986944 x^6-2099249152 x^7+66913566720 x^8+2 \left (-588408391+33415183048 x+992171122 x^2+48280731648 x^3+3284140032 x^4+5275074560 x^5+565182464 x^6+984023040 x^7\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (2167822549+88760534448 x+3694527828 x^2+308588576768 x^3-10745282560 x^4+214715203584 x^5-2099249152 x^6+66913566720 x^7+4 \left (5472731492-218977735 x+23006101504 x^2+1500774400 x^3+2391531520 x^4+282591232 x^5+492011520 x^6\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{39729930240 \left (x+\sqrt {1+x^2}\right )^{7/2}}-\frac {545 \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{8192} \]

[In]

Integrate[(1 + x^2)^(3/2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]

[Out]

(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(15903121112 + 5227043711*x + 220397520304*x^2 - 1415707308*x^3 + 407581982
720*x^4 - 11794907136*x^5 + 248171986944*x^6 - 2099249152*x^7 + 66913566720*x^8 + 2*(-588408391 + 33415183048*
x + 992171122*x^2 + 48280731648*x^3 + 3284140032*x^4 + 5275074560*x^5 + 565182464*x^6 + 984023040*x^7)*Sqrt[x
+ Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(2167822549 + 88760534448*x + 3694527828*x^2 + 308588576768*x^3 - 10745282560
*x^4 + 214715203584*x^5 - 2099249152*x^6 + 66913566720*x^7 + 4*(5472731492 - 218977735*x + 23006101504*x^2 + 1
500774400*x^3 + 2391531520*x^4 + 282591232*x^5 + 492011520*x^6)*Sqrt[x + Sqrt[1 + x^2]])))/(39729930240*(x + S
qrt[1 + x^2])^(7/2)) - (545*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/8192

Maple [F]

\[\int \left (x^{2}+1\right )^{\frac {3}{2}} \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}d x\]

[In]

int((x^2+1)^(3/2)*(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)^(3/2)*(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 = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.50 \[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{39729930240} \, {\left (246005760 \, x^{4} + 377783296 \, x^{3} + 987937568 \, x^{2} + 2 \, {\left (123002880 \, x^{3} - 47596032 \, x^{2} + 578794096 \, x - 588408391\right )} \sqrt {x^{2} + 1} - {\left (1493606400 \, x^{4} + 391339520 \, x^{3} + 7419648592 \, x^{2} - {\left (9857802240 \, x^{3} + 128933376 \, x^{2} + 25148050000 \, x + 2167822549\right )} \sqrt {x^{2} + 1} + 3444246485 \, x - 15903121112\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 2654539406 \, x + 21890925968\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {545}{16384} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {545}{16384} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \]

[In]

integrate((x^2+1)^(3/2)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

1/39729930240*(246005760*x^4 + 377783296*x^3 + 987937568*x^2 + 2*(123002880*x^3 - 47596032*x^2 + 578794096*x -
588408391)*sqrt(x^2 + 1) - (1493606400*x^4 + 391339520*x^3 + 7419648592*x^2 - (9857802240*x^3 + 128933376*x^2
+ 25148050000*x + 2167822549)*sqrt(x^2 + 1) + 3444246485*x - 15903121112)*sqrt(x + sqrt(x^2 + 1)) + 265453940
6*x + 21890925968)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 545/16384*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) +
545/16384*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1)

Sympy [F]

\[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )^{\frac {3}{2}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\, dx \]

[In]

integrate((x**2+1)**(3/2)*(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))*(x**2 + 1)**(3/2)*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1), x)

Maxima [F]

\[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { {\left (x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} \,d x } \]

[In]

integrate((x^2+1)^(3/2)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((x^2 + 1)^(3/2)*sqrt(x + sqrt(x^2 + 1))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1), x)

Giac [F(-1)]

Timed out. \[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Timed out} \]

[In]

integrate((x^2+1)^(3/2)*(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. \[ \int \left (1+x^2\right )^{3/2} \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,{\left (x^2+1\right )}^{3/2}\,\sqrt {x+\sqrt {x^2+1}} \,d x \]

[In]

int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(3/2)*(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)^(3/2)*(x + (x^2 + 1)^(1/2))^(1/2), x)