3.28.65 \(\int \frac {(1+x^2)^2}{(-1+x^2)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [2765]

3.28.65.1 Optimal result

Integrand size = 32, antiderivative size = 261 \[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x \left (-5+x^2\right )}{2 \left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}-4 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2 \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

output

1/2*x*(x^2-5)/(x^2-1)/(x^2+(x^4+1)^(1/2))^(1/2)-4*arctan(2^(1/2)*x*(x^2+(x 
^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)+(14+10*2^(1/2))^(1/2)*ar 
ctan((-2+2*2^(1/2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2) 
))+1/2*arctanh(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))* 
2^(1/2)-(-14+10*2^(1/2))^(1/2)*arctanh((2+2*2^(1/2))^(1/2)*x*(x^2+(x^4+1)^ 
(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))
 

3.28.65.2 Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x \left (-5+x^2\right )}{2 \left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}-4 \sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2 \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]

input

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

output

(x*(-5 + x^2))/(2*(-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) - 4*Sqrt[2]*ArcTan 
[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[ 
2*(7 + 5*Sqrt[2])]*ArcTan[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2*(1 + Sqrt[2]) 
]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] + ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt 
[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])]/Sqrt[2] - Sqrt[2*(-7 + 5*Sqrt[2])]*ArcTa 
nh[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x^4]))/(x*Sqrt[x^2 + Sqrt[1 
 + x^4]])]
 

3.28.65.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.28.65.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.28.65.4 Maple [F]

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

input

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

output

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

3.28.65.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (204) = 408\).

Time = 7.02 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.52 \[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {16 \, \sqrt {2} {\left (x^{2} - 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) + \sqrt {2} {\left (x^{2} - 1\right )} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + 2 \, {\left (x^{2} - 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (-\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} + {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} - 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} - 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) - 2 \, {\left (x^{2} - 1\right )} \sqrt {10 \, \sqrt {2} - 14} \log \left (-\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} - {\left (4 \, x^{3} + \sqrt {2} {\left (3 \, x^{3} - 7 \, x\right )} - \sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} - 10 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {10 \, \sqrt {2} - 14} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) - 2 \, {\left (x^{2} - 1\right )} \sqrt {-10 \, \sqrt {2} - 14} \log \left (\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x - 4 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14} + {\left (4 \, x^{3} - \sqrt {2} {\left (3 \, x^{3} - 7 \, x\right )} - 10 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} - 1}\right ) + 2 \, {\left (x^{2} - 1\right )} \sqrt {-10 \, \sqrt {2} - 14} \log \left (\frac {2 \, \sqrt {2} x^{2} - 4 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (3 \, \sqrt {2} x - 4 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14} + {\left (4 \, x^{3} - \sqrt {2} {\left (3 \, x^{3} - 7 \, x\right )} - 10 \, x\right )} \sqrt {-10 \, \sqrt {2} - 14}\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} - 1}\right ) - 4 \, {\left (x^{5} - 5 \, x^{3} - \sqrt {x^{4} + 1} {\left (x^{3} - 5 \, x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{2} - 1\right )}} \]

input

integrate((x^2+1)^2/(x^2-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fric 
as")
 

output

1/8*(16*sqrt(2)*(x^2 - 1)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 + 1) 
)*sqrt(x^2 + sqrt(x^4 + 1))/x) + sqrt(2)*(x^2 - 1)*log(4*x^4 + 4*sqrt(x^4 
+ 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 
 1)) + 1) + 2*(x^2 - 1)*sqrt(10*sqrt(2) - 14)*log(-(2*sqrt(2)*x^2 + 4*x^2 
+ (4*x^3 + sqrt(2)*(3*x^3 - 7*x) - sqrt(x^4 + 1)*(3*sqrt(2)*x + 4*x) - 10* 
x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(10*sqrt(2) - 14) + 2*sqrt(x^4 + 1)*(sqrt 
(2) + 1))/(x^2 - 1)) - 2*(x^2 - 1)*sqrt(10*sqrt(2) - 14)*log(-(2*sqrt(2)*x 
^2 + 4*x^2 - (4*x^3 + sqrt(2)*(3*x^3 - 7*x) - sqrt(x^4 + 1)*(3*sqrt(2)*x + 
 4*x) - 10*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(10*sqrt(2) - 14) + 2*sqrt(x^4 
 + 1)*(sqrt(2) + 1))/(x^2 - 1)) - 2*(x^2 - 1)*sqrt(-10*sqrt(2) - 14)*log(( 
2*sqrt(2)*x^2 - 4*x^2 + sqrt(x^2 + sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(3*sqrt(2 
)*x - 4*x)*sqrt(-10*sqrt(2) - 14) + (4*x^3 - sqrt(2)*(3*x^3 - 7*x) - 10*x) 
*sqrt(-10*sqrt(2) - 14)) + 2*sqrt(x^4 + 1)*(sqrt(2) - 1))/(x^2 - 1)) + 2*( 
x^2 - 1)*sqrt(-10*sqrt(2) - 14)*log((2*sqrt(2)*x^2 - 4*x^2 - sqrt(x^2 + sq 
rt(x^4 + 1))*(sqrt(x^4 + 1)*(3*sqrt(2)*x - 4*x)*sqrt(-10*sqrt(2) - 14) + ( 
4*x^3 - sqrt(2)*(3*x^3 - 7*x) - 10*x)*sqrt(-10*sqrt(2) - 14)) + 2*sqrt(x^4 
 + 1)*(sqrt(2) - 1))/(x^2 - 1)) - 4*(x^5 - 5*x^3 - sqrt(x^4 + 1)*(x^3 - 5* 
x))*sqrt(x^2 + sqrt(x^4 + 1)))/(x^2 - 1)
 

3.28.65.6 Sympy [F]

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

input

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

output

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

3.28.65.7 Maxima [F]

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

input

integrate((x^2+1)^2/(x^2-1)^2/(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxi 
ma")
 

output

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

3.28.65.8 Giac [F]

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

input

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

output

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

3.28.65.9 Mupad [F(-1)]

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

input

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

output

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