3.31.41 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2)^{5/2}} \, dx\) [3041]

3.31.41.1 Optimal result

Integrand size = 31, antiderivative size = 452 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx=\frac {\left (-117-7 x-151 x^2-27 x^3+30 x^4-20 x^5\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (10 x+8 x^2+34 x^3+8 x^4+24 x^5\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-1-166 x-17 x^2+30 x^3-20 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (2+4 x+22 x^2+8 x^3+24 x^4\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{192 \sqrt {1+x^2} \left (3 x+7 x^3+4 x^5\right )+192 \left (1+6 x^2+9 x^4+4 x^6\right )}+\frac {1}{256} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {40 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-7 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]

output

Unintegrable
 

3.31.41.2 Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx=\frac {1}{768} \left (\frac {4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-117-7 x-151 x^2-27 x^3+30 x^4-20 x^5+2 x \left (5+4 x+17 x^2+4 x^3+12 x^4\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-1-166 x-17 x^2+30 x^3-20 x^4+\left (2+4 x+22 x^2+8 x^3+24 x^4\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{1+6 x^2+9 x^4+4 x^6+x \sqrt {1+x^2} \left (3+7 x^2+4 x^4\right )}+384 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {10 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+69 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {40 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+40 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-25 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-72 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {90 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+70 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-45 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+11 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \]

input

Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2),x]
 

output

((4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-117 - 7*x - 151*x^2 - 27*x^3 + 30* 
x^4 - 20*x^5 + 2*x*(5 + 4*x + 17*x^2 + 4*x^3 + 12*x^4)*Sqrt[x + Sqrt[1 + x 
^2]] + Sqrt[1 + x^2]*(-1 - 166*x - 17*x^2 + 30*x^3 - 20*x^4 + (2 + 4*x + 2 
2*x^2 + 8*x^3 + 24*x^4)*Sqrt[x + Sqrt[1 + x^2]])))/(1 + 6*x^2 + 9*x^4 + 4* 
x^6 + x*Sqrt[1 + x^2]*(3 + 7*x^2 + 4*x^4)) + 384*RootSum[2 - 4*#1^2 + 6*#1 
^4 - 4*#1^6 + #1^8 & , (10*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 6 
*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x 
+ Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] 
*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ] + 69*RootSum[2 - 4*#1^2 + 6*#1^4 
 - 4*#1^6 + #1^8 & , (40*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 40* 
Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 25*Log[Sqrt[1 + Sqrt[x 
+ Sqrt[1 + x^2]]] - #1]*#1^4 + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - # 
1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ] - 72*RootSum[2 - 4*#1^2 + 6*#1 
^4 - 4*#1^6 + #1^8 & , (90*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 7 
0*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 45*Log[Sqrt[1 + Sqrt[ 
x + Sqrt[1 + x^2]]] - #1]*#1^4 + 11*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] 
- #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/768
 

3.31.41.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (x^2+1\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (x^2+1\right )^{5/2}}dx\)

input

Int[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2),x]
 

output

$Aborted
 

3.31.41.3.1 Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.31.41.4 Maple [N/A] (verified)

Not integrable

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.05

\[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{\frac {5}{2}}}d x\]

input

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(5/2),x)
 

output

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(5/2),x)
 

3.31.41.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.02 (sec) , antiderivative size = 3420, normalized size of antiderivative = 7.57 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

input

integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(5/2),x, algorithm="fr 
icas")
 

output

1/768*(3*sqrt(2)*(x^4 + 2*x^2 + 1)*sqrt(sqrt(2)*sqrt(-402653184*(131/65536 
*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 9 
63/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456* 
I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 3276 
8*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*I*sqr 
t(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) 
+ 126153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/10 
73741824) + 548597/4) + 8192*sqrt(487277/268435456*I*sqrt(2) + 582919/1073 
741824) + 8192*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963 
/2)*log(1/4*(2147483648*(29673954056209*sqrt(2)*(131*I*sqrt(2) + 32768*sqr 
t(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 243212876973803 
14*sqrt(2))*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 
582919/1073741824) - 963/65536)^2 + 52229567628427796752105472*sqrt(2)*(-1 
31/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/10737418 
24) - 963/65536)^2 + (63724331107212100370432*sqrt(2)*(-131/65536*I*sqrt(2 
) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^ 
2 - 57152035512258534*sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456 
*I*sqrt(2) + 582919/1073741824) + 963) + 17015803710063315259*sqrt(2))*(-1 
31*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) 
 + 963) - 4*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268...
 

3.31.41.6 Sympy [N/A]

Not integrable

Time = 11.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.06 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x^{2} + 1\right )^{\frac {5}{2}}}\, dx \]

input

integrate((1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(x**2+1)**(5/2),x)
 

output

Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2 + 1)**(5/2), x)
 

3.31.41.7 Maxima [N/A]

Not integrable

Time = 0.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.06 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(5/2),x, algorithm="ma 
xima")
 

output

integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^(5/2), x)
 

3.31.41.8 Giac [N/A]

Not integrable

Time = 126.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.06 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(5/2),x, algorithm="gi 
ac")
 

output

integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^(5/2), x)
 

3.31.41.9 Mupad [N/A]

Not integrable

Time = 7.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.06 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{{\left (x^2+1\right )}^{5/2}} \,d x \]

input

int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/(x^2 + 1)^(5/2),x)
 

output

int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/(x^2 + 1)^(5/2), x)