3.26.89 \(\int \frac {(1+x^2)^2}{(-1+x^2)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [2589]

3.26.89.1 Optimal result

Integrand size = 41, antiderivative size = 224 \[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {-x^2 \left (-1+x^2\right )-x^2 \sqrt {1+x^4}}{x \left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

output

(-x^2*(x^2-1)-(x^4+1)^(1/2)*x^2)/x/(x^2-1)/(x^2+(x^4+1)^(1/2))^(1/2)+arcta 
n(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)-(1+2^ 
(1/2))^(1/2)*arctan((-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)))+(2^(1/2)-1)^(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)))
 

3.26.89.2 Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {x \left (-1+x^2+\sqrt {1+x^4}\right )}{\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]

input

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

output

-((x*(-1 + x^2 + Sqrt[1 + x^4]))/((-1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]])) + 
 Sqrt[2]*ArcTan[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + 
x^4]])] - Sqrt[1 + Sqrt[2]]*ArcTan[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2*(1 + 
 Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] + Sqrt[-1 + Sqrt[2]]*ArcTanh[(Sqr 
t[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x^4]))/(x*Sqrt[x^2 + Sqrt[1 + x^4] 
])]
 

3.26.89.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.26.89.3.1 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] 
]
 

3.26.89.4 Maple [F]

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

input

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

output

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

3.26.89.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (179) = 358\).

Time = 1.24 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.42 \[ \int \frac {\left (1+x^2\right )^2}{\left (-1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {2 \, \sqrt {2} {\left (x^{2} - 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) + {\left (x^{2} - 1\right )} \sqrt {\sqrt {2} - 1} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} {\left (x^{3} - 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} - 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) - {\left (x^{2} - 1\right )} \sqrt {\sqrt {2} - 1} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} {\left (x^{3} - 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} - 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) - {\left (x^{2} - 1\right )} \sqrt {-\sqrt {2} - 1} \log \left (\frac {\sqrt {2} x^{2} - 2 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )} \sqrt {-\sqrt {2} - 1} + {\left (x^{3} - \sqrt {2} {\left (x^{3} - 2 \, x\right )} - 3 \, x\right )} \sqrt {-\sqrt {2} - 1}\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} - 1}\right ) + {\left (x^{2} - 1\right )} \sqrt {-\sqrt {2} - 1} \log \left (\frac {\sqrt {2} x^{2} - 2 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )} \sqrt {-\sqrt {2} - 1} + {\left (x^{3} - \sqrt {2} {\left (x^{3} - 2 \, x\right )} - 3 \, x\right )} \sqrt {-\sqrt {2} - 1}\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} - 1}\right ) + 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, {\left (x^{2} - 1\right )}} \]

input

integrate((x^2+1)^2/(x^2-1)^2/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, a 
lgorithm="fricas")
 

output

-1/4*(2*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) + (x^2 - 1)*sqrt(sqrt(2) - 1)*log(-(sqrt(2) 
*x^2 + 2*x^2 + (x^3 + sqrt(2)*(x^3 - 2*x) - sqrt(x^4 + 1)*(sqrt(2)*x + x) 
- 3*x)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) - 1) + sqrt(x^4 + 1)*(sqrt(2 
) + 1))/(x^2 - 1)) - (x^2 - 1)*sqrt(sqrt(2) - 1)*log(-(sqrt(2)*x^2 + 2*x^2 
 - (x^3 + sqrt(2)*(x^3 - 2*x) - sqrt(x^4 + 1)*(sqrt(2)*x + x) - 3*x)*sqrt( 
x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) - 1) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 
 - 1)) - (x^2 - 1)*sqrt(-sqrt(2) - 1)*log((sqrt(2)*x^2 - 2*x^2 + sqrt(x^2 
+ sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(sqrt(2)*x - x)*sqrt(-sqrt(2) - 1) + (x^3 
- sqrt(2)*(x^3 - 2*x) - 3*x)*sqrt(-sqrt(2) - 1)) + sqrt(x^4 + 1)*(sqrt(2) 
- 1))/(x^2 - 1)) + (x^2 - 1)*sqrt(-sqrt(2) - 1)*log((sqrt(2)*x^2 - 2*x^2 - 
 sqrt(x^2 + sqrt(x^4 + 1))*(sqrt(x^4 + 1)*(sqrt(2)*x - x)*sqrt(-sqrt(2) - 
1) + (x^3 - sqrt(2)*(x^3 - 2*x) - 3*x)*sqrt(-sqrt(2) - 1)) + sqrt(x^4 + 1) 
*(sqrt(2) - 1))/(x^2 - 1)) + 4*(x^3 - sqrt(x^4 + 1)*x + x)*sqrt(x^2 + sqrt 
(x^4 + 1)))/(x^2 - 1)
 

3.26.89.6 Sympy [F]

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

input

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

3.26.89.7 Maxima [F]

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

input

integrate((x^2+1)^2/(x^2-1)^2/(x^4+1)^(1/2)/(x^2+(x^4+1)^(1/2))^(1/2),x, a 
lgorithm="maxima")
 

output

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

3.26.89.8 Giac [F]

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

input

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

output

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

3.26.89.9 Mupad [F(-1)]

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

input

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