\(\int \frac {\sqrt {1+x^2} (1+x^4) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx\) [2972]

Optimal result

Integrand size = 45, antiderivative size = 375 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\frac {\left (-78032+1254 x-193024 x^2+3072 x^3-35840 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-184-345 x-2048 x^2-2560 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-282-175104 x+3072 x^2-35840 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (935-2048 x-2560 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{80640 x \sqrt {1+x^2}+40320 \left (1+2 x^2\right )}+\frac {251}{128} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \] Output:

Unintegrable
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (2 \left (39016-627 x+96512 x^2-1536 x^3+17920 x^4\right )+\left (184+345 x+2048 x^2+2560 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (282+175104 x-3072 x^2+35840 x^3+\left (-935+2048 x+2560 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{40320 \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}+\frac {251}{128} \text {arctanh}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \] Input:

Integrate[(Sqrt[1 + x^2]*(1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - 
 x^4),x]
 

Output:

-1/40320*(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(2*(39016 - 627*x + 96512*x^2 
- 1536*x^3 + 17920*x^4) + (184 + 345*x + 2048*x^2 + 2560*x^3)*Sqrt[x + Sqr 
t[1 + x^2]] + Sqrt[1 + x^2]*(282 + 175104*x - 3072*x^2 + 35840*x^3 + (-935 
 + 2048*x + 2560*x^2)*Sqrt[x + Sqrt[1 + x^2]])))/(1 + 2*x^2 + 2*x*Sqrt[1 + 
 x^2]) + (251*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/128 - RootSum[-2 
 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/ 
(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[ 
Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]
 

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.

\(\displaystyle \int \frac {\sqrt {x^2+1} \left (x^4+1\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{1-x^4} \, dx\)

\(\Big \downarrow \) 1388

\(\displaystyle \int \frac {\left (x^4+1\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\left (1-x^2\right ) \sqrt {x^2+1}}dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\sqrt {x^2+1}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(1-x) \sqrt {x^2+1}}dx-\int \frac {x^2 \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\sqrt {x^2+1}}dx+\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{(x+1) \sqrt {x^2+1}}dx\)

Input:

Int[(Sqrt[1 + x^2]*(1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4), 
x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.10

\[\int \frac {\sqrt {x^{2}+1}\, \left (x^{4}+1\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{-x^{4}+1}d x\]

Input:

int((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x)
 

Output:

int((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x)
 

Fricas [C] (verification not implemented)

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

Time = 0.18 (sec) , antiderivative size = 974, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\text {Too large to display} \] Input:

integrate((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1) 
,x, algorithm="fricas")
 

Output:

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 - 78 
032)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1/2*sqrt(2*sqrt(sqrt(2) + 1) + 2) 
*log(sqrt(2)*sqrt(2*sqrt(sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1) 
) + 1)) - 1/2*sqrt(2*sqrt(sqrt(2) + 1) + 2)*log(-sqrt(2)*sqrt(2*sqrt(sqrt( 
2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(sqr 
t(2) + 1) + 2)*log(sqrt(2)*sqrt(-2*sqrt(sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x 
+ sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(sqrt(2) + 1) + 2)*log(-sqrt(2)*s 
qrt(-2*sqrt(sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2 
*sqrt(2*sqrt(sqrt(2) - 1) + 2)*log(sqrt(2)*sqrt(2*sqrt(sqrt(2) - 1) + 2) + 
 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(2*sqrt(sqrt(2) - 1) + 2)* 
log(-sqrt(2)*sqrt(2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1) 
) + 1)) - 1/2*sqrt(-2*sqrt(sqrt(2) - 1) + 2)*log(sqrt(2)*sqrt(-2*sqrt(sqrt 
(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(sq 
rt(2) - 1) + 2)*log(-sqrt(2)*sqrt(-2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt( 
x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(-sqrt(2) + 1) + 2)*log(sqrt(2)* 
sqrt(2*sqrt(-sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/ 
2*sqrt(2*sqrt(-sqrt(2) + 1) + 2)*log(-sqrt(2)*sqrt(2*sqrt(-sqrt(2) + 1) + 
2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(-sqrt(2) + 1) 
 + 2)*log(sqrt(2)*sqrt(-2*sqrt(-sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqr...
 

Sympy [N/A]

Not integrable

Time = 33.99 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=- \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx - \int \frac {x^{4} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx \] Input:

integrate((x**2+1)**(1/2)*(x**4+1)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/( 
-x**4+1),x)
 

Output:

-Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2*sqrt(x**2 + 1) - sqrt(x 
**2 + 1)), x) - Integral(x**4*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2*sqr 
t(x**2 + 1) - sqrt(x**2 + 1)), x)
 

Maxima [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\int { -\frac {{\left (x^{4} + 1\right )} \sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1} \,d x } \] Input:

integrate((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1) 
,x, algorithm="maxima")
 

Output:

-integrate((x^4 + 1)*sqrt(x^2 + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^4 
- 1), x)
 

Giac [N/A]

Not integrable

Time = 175.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\int { -\frac {{\left (x^{4} + 1\right )} \sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1} \,d x } \] Input:

integrate((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1) 
,x, algorithm="giac")
 

Output:

integrate(-(x^4 + 1)*sqrt(x^2 + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^4 
- 1), x)
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=\int -\frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x^2+1}\,\left (x^4+1\right )}{x^4-1} \,d x \] Input:

int(-(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(1/2)*(x^4 + 1))/( 
x^4 - 1),x)
 

Output:

int(-(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(1/2)*(x^4 + 1))/( 
x^4 - 1), x)
 

Reduce [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx=-\left (\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x \right )-\left (\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, x^{6}}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x \right )-\left (\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, x^{4}}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x \right )-\left (\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, x^{2}}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x \right )-\left (\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, \sqrt {x^{2}+1}\, x^{5}}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x \right )+\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, \sqrt {x^{2}+1}\, x^{4}}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x -\left (\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, \sqrt {x^{2}+1}\, x}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x \right )+\int \frac {\sqrt {\sqrt {\sqrt {x^{2}+1}+x}+1}\, \sqrt {x^{2}+1}}{\sqrt {x^{2}+1}\, x^{4}-\sqrt {x^{2}+1}+x^{5}-x^{4}-x +1}d x \] Input:

int((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x)
 

Output:

 - int(sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)/(sqrt(x**2 + 1)*x**4 - sqrt(x**2 
 + 1) + x**5 - x**4 - x + 1),x) - int((sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)* 
x**6)/(sqrt(x**2 + 1)*x**4 - sqrt(x**2 + 1) + x**5 - x**4 - x + 1),x) - in 
t((sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)*x**4)/(sqrt(x**2 + 1)*x**4 - sqrt(x* 
*2 + 1) + x**5 - x**4 - x + 1),x) - int((sqrt(sqrt(sqrt(x**2 + 1) + x) + 1 
)*x**2)/(sqrt(x**2 + 1)*x**4 - sqrt(x**2 + 1) + x**5 - x**4 - x + 1),x) - 
int((sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)*sqrt(x**2 + 1)*x**5)/(sqrt(x**2 + 
1)*x**4 - sqrt(x**2 + 1) + x**5 - x**4 - x + 1),x) + int((sqrt(sqrt(sqrt(x 
**2 + 1) + x) + 1)*sqrt(x**2 + 1)*x**4)/(sqrt(x**2 + 1)*x**4 - sqrt(x**2 + 
 1) + x**5 - x**4 - x + 1),x) - int((sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)*sq 
rt(x**2 + 1)*x)/(sqrt(x**2 + 1)*x**4 - sqrt(x**2 + 1) + x**5 - x**4 - x + 
1),x) + int((sqrt(sqrt(sqrt(x**2 + 1) + x) + 1)*sqrt(x**2 + 1))/(sqrt(x**2 
 + 1)*x**4 - sqrt(x**2 + 1) + x**5 - x**4 - x + 1),x)