3.30.60 \(\int \frac {(1+x^4)^2}{(-1+x^4)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [2960]

3.30.60.1 Optimal result

Integrand size = 32, antiderivative size = 362 \[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x \left (-3+x^4\right )}{2 \left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}}-\frac {1}{2} \sqrt {\frac {1}{2} \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

output

1/2*x*(x^4-3)/(x^4-1)/(x^2+(x^4+1)^(1/2))^(1/2)+1/4*(14+10*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)))-1/4*(14+10*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)))+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)-1/4*(-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)))-1 
/4*(-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)))
 

3.30.60.2 Mathematica [A] (verified)

Time = 4.37 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {1}{4} \left (\frac {2 x \left (-3+x^4\right )}{\left (-1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\sqrt {2 \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2 \left (7+5 \sqrt {2}\right )} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2 \left (-7+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \]

input

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

output

((2*x*(-3 + x^4))/((-1 + x^4)*Sqrt[x^2 + Sqrt[1 + x^4]]) - Sqrt[2*(7 + 5*S 
qrt[2])]*ArcTan[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x^4]))/(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]])] + 2*Sqrt[2] 
*ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])] 
 - Sqrt[2*(-7 + 5*Sqrt[2])]*ArcTanh[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqr 
t[1 + x^4]))/(x*Sqrt[x^2 + Sqrt[1 + x^4]])] - Sqrt[2*(-7 + 5*Sqrt[2])]*Arc 
Tanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqrt[2*(1 + Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 
 + x^4]])])/4
 

3.30.60.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.30.60.3.1 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] 
]
 

3.30.60.4 Maple [F]

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

input

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

output

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

3.30.60.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (267) = 534\).

Time = 4.76 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.75 \[ \int \frac {\left (1+x^4\right )^2}{\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {\sqrt {2} {\left (x^{4} - 1\right )} \sqrt {-5 \, \sqrt {2} - 7} \log \left (-\frac {2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} - 7 \, x^{2}\right )} \sqrt {-5 \, \sqrt {2} - 7} + 2 \, {\left (\sqrt {2} x^{3} - 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (17 \, x^{4} - 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 3\right )} \sqrt {-5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) - \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {-5 \, \sqrt {2} - 7} \log \left (\frac {2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} - 7 \, x^{2}\right )} \sqrt {-5 \, \sqrt {2} - 7} - 2 \, {\left (\sqrt {2} x^{3} - 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (17 \, x^{4} - 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 3\right )} \sqrt {-5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) - \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} - 7} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (17 \, x^{4} + 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} + 7 \, x^{2}\right )} + 3\right )} \sqrt {5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) + \sqrt {2} {\left (x^{4} - 1\right )} \sqrt {5 \, \sqrt {2} - 7} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (17 \, x^{4} + 2 \, \sqrt {2} {\left (6 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (5 \, \sqrt {2} x^{2} + 7 \, x^{2}\right )} + 3\right )} \sqrt {5 \, \sqrt {2} - 7}}{x^{4} - 1}\right ) + 2 \, \sqrt {2} {\left (x^{4} - 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 ) - 8 \, {\left (x^{7} - 3 \, x^{3} - {\left (x^{5} - 3 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{16 \, {\left (x^{4} - 1\right )}} \]

input

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

output

1/16*(sqrt(2)*(x^4 - 1)*sqrt(-5*sqrt(2) - 7)*log(-(2*sqrt(x^4 + 1)*(5*sqrt 
(2)*x^2 - 7*x^2)*sqrt(-5*sqrt(2) - 7) + 2*(sqrt(2)*x^3 - 2*x^3 + sqrt(x^4 
+ 1)*(sqrt(2)*x - x))*sqrt(x^2 + sqrt(x^4 + 1)) - (17*x^4 - 2*sqrt(2)*(6*x 
^4 + 1) + 3)*sqrt(-5*sqrt(2) - 7))/(x^4 - 1)) - sqrt(2)*(x^4 - 1)*sqrt(-5* 
sqrt(2) - 7)*log((2*sqrt(x^4 + 1)*(5*sqrt(2)*x^2 - 7*x^2)*sqrt(-5*sqrt(2) 
- 7) - 2*(sqrt(2)*x^3 - 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x - x))*sqrt(x^2 + 
sqrt(x^4 + 1)) - (17*x^4 - 2*sqrt(2)*(6*x^4 + 1) + 3)*sqrt(-5*sqrt(2) - 7) 
)/(x^4 - 1)) - sqrt(2)*(x^4 - 1)*sqrt(5*sqrt(2) - 7)*log((2*(sqrt(2)*x^3 + 
 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x + x))*sqrt(x^2 + sqrt(x^4 + 1)) + (17*x^ 
4 + 2*sqrt(2)*(6*x^4 + 1) + 2*sqrt(x^4 + 1)*(5*sqrt(2)*x^2 + 7*x^2) + 3)*s 
qrt(5*sqrt(2) - 7))/(x^4 - 1)) + sqrt(2)*(x^4 - 1)*sqrt(5*sqrt(2) - 7)*log 
((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x + x))*sqrt(x^2 + sqrt( 
x^4 + 1)) - (17*x^4 + 2*sqrt(2)*(6*x^4 + 1) + 2*sqrt(x^4 + 1)*(5*sqrt(2)*x 
^2 + 7*x^2) + 3)*sqrt(5*sqrt(2) - 7))/(x^4 - 1)) + 2*sqrt(2)*(x^4 - 1)*log 
(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*s 
qrt(x^2 + sqrt(x^4 + 1)) + 1) - 8*(x^7 - 3*x^3 - (x^5 - 3*x)*sqrt(x^4 + 1) 
)*sqrt(x^2 + sqrt(x^4 + 1)))/(x^4 - 1)
 

3.30.60.6 Sympy [F]

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

input

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

output

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

3.30.60.7 Maxima [F]

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

input

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

output

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

3.30.60.8 Giac [F]

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

input

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

output

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

3.30.60.9 Mupad [F(-1)]

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

input

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

output

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