\(\int \frac {(-1+x^4)^2}{(1+x^4)^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [2374]

Optimal result

Integrand size = 32, antiderivative size = 189 \[ \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x^2 \sqrt {1+x^4} \left (9 x^2+6 x^6\right )+x^2 \left (4+12 x^4+6 x^8\right )}{2 x \sqrt {1+x^4} \left (2 x^2+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+2 x \left (1+3 x^4+2 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \] Output:

(x^2*(x^4+1)^(1/2)*(6*x^6+9*x^2)+x^2*(6*x^8+12*x^4+4))/(2*x*(x^4+1)^(1/2)* 
(2*x^6+2*x^2)*(x^2+(x^4+1)^(1/2))^(1/2)+2*x*(2*x^8+3*x^4+1)*(x^2+(x^4+1)^( 
1/2))^(1/2))-3/2*arctanh(x/(x^2+(x^4+1)^(1/2))^(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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {1}{2} \left (\frac {x \left (4+12 x^4+6 x^8+9 x^2 \sqrt {1+x^4}+6 x^6 \sqrt {1+x^4}\right )}{\left (1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}} \left (1+2 x^4+2 x^2 \sqrt {1+x^4}\right )}-3 \text {arctanh}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \] Input:

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

Output:

((x*(4 + 12*x^4 + 6*x^8 + 9*x^2*Sqrt[1 + x^4] + 6*x^6*Sqrt[1 + x^4]))/((1 
+ x^4)*Sqrt[x^2 + Sqrt[1 + x^4]]*(1 + 2*x^4 + 2*x^2*Sqrt[1 + x^4])) - 3*Ar 
cTanh[x/Sqrt[x^2 + Sqrt[1 + x^4]]] + Sqrt[2]*ArcTanh[(-1 + x^2 + Sqrt[1 + 
x^4])/(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])])/2
 

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

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

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{\sqrt {x^2+\sqrt {x^4+1}}}dx+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {x^4+1}}}dx+\frac {3}{4} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} i \int \frac {1}{\left (-x-(-1)^{3/4}\right )^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {3}{4} (-1)^{3/4} \int \frac {1}{\left (-x-(-1)^{3/4}\right ) \sqrt {x^2+\sqrt {x^4+1}}}dx+\frac {1}{4} i \int \frac {1}{\left (x+\sqrt [4]{-1}\right )^2 \sqrt {x^2+\sqrt {x^4+1}}}dx+\frac {3}{4} \sqrt [4]{-1} \int \frac {1}{\left (x+\sqrt [4]{-1}\right ) \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {1}{4} i \int \frac {1}{\left (x-(-1)^{3/4}\right )^2 \sqrt {x^2+\sqrt {x^4+1}}}dx-\frac {3}{4} (-1)^{3/4} \int \frac {1}{\left (x-(-1)^{3/4}\right ) \sqrt {x^2+\sqrt {x^4+1}}}dx-4 \text {arctanh}\left (\frac {x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )\)

Input:

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

Output:

$Aborted
 

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)
 

Fricas [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {\sqrt {2} {\left (x^{4} + 1\right )} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + 3 \, {\left (x^{4} + 1\right )} \log \left (-\frac {9 \, x^{4} + 8 \, \sqrt {x^{4} + 1} x^{2} - 4 \, {\left (2 \, x^{3} + \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1}{x^{4} + 1}\right ) - 4 \, {\left (x^{7} + 3 \, x^{3} - {\left (x^{5} + 4 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{4} + 1\right )}} \] Input:

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

Output:

1/8*(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)*sqrt(x^2 + sqrt(x^4 + 1)) + 1) + 3*(x^4 + 1)*log( 
-(9*x^4 + 8*sqrt(x^4 + 1)*x^2 - 4*(2*x^3 + sqrt(x^4 + 1)*x)*sqrt(x^2 + sqr 
t(x^4 + 1)) + 1)/(x^4 + 1)) - 4*(x^7 + 3*x^3 - (x^5 + 4*x)*sqrt(x^4 + 1))* 
sqrt(x^2 + sqrt(x^4 + 1)))/(x^4 + 1)
 

Sympy [F]

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

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

Output:

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

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)
 

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)
 

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)
 

Reduce [F]

\[ \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {8 \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{5}+100 \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x -3 \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-\sqrt {2}\, x \right ) x^{4}-3 \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-\sqrt {2}\, x \right )+3 \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+\sqrt {2}\, x \right ) x^{4}+3 \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+\sqrt {2}\, x \right )-32 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{10}}{x^{8}+2 x^{4}+1}d x \right ) x^{4}-32 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{10}}{x^{8}+2 x^{4}+1}d x \right )-60 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{6}}{x^{8}+2 x^{4}+1}d x \right ) x^{4}-60 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{6}}{x^{8}+2 x^{4}+1}d x \right )-124 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}}{x^{8}+2 x^{4}+1}d x \right ) x^{4}-124 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}}{x^{8}+2 x^{4}+1}d x \right )-88 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{8}+2 x^{4}+1}d x \right ) x^{4}-88 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}}{x^{8}+2 x^{4}+1}d x \right )}{24 x^{4}+24} \] Input:

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

Output:

(8*sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)*x**5 + 100*sqrt(sqrt(x**4 + 
1) + x**2)*sqrt(x**4 + 1)*x - 3*sqrt(2)*log(sqrt(sqrt(x**4 + 1) + x**2) - 
sqrt(2)*x)*x**4 - 3*sqrt(2)*log(sqrt(sqrt(x**4 + 1) + x**2) - sqrt(2)*x) + 
 3*sqrt(2)*log(sqrt(sqrt(x**4 + 1) + x**2) + sqrt(2)*x)*x**4 + 3*sqrt(2)*l 
og(sqrt(sqrt(x**4 + 1) + x**2) + sqrt(2)*x) - 32*int((sqrt(sqrt(x**4 + 1) 
+ x**2)*x**10)/(x**8 + 2*x**4 + 1),x)*x**4 - 32*int((sqrt(sqrt(x**4 + 1) + 
 x**2)*x**10)/(x**8 + 2*x**4 + 1),x) - 60*int((sqrt(sqrt(x**4 + 1) + x**2) 
*x**6)/(x**8 + 2*x**4 + 1),x)*x**4 - 60*int((sqrt(sqrt(x**4 + 1) + x**2)*x 
**6)/(x**8 + 2*x**4 + 1),x) - 124*int((sqrt(sqrt(x**4 + 1) + x**2)*x**2)/( 
x**8 + 2*x**4 + 1),x)*x**4 - 124*int((sqrt(sqrt(x**4 + 1) + x**2)*x**2)/(x 
**8 + 2*x**4 + 1),x) - 88*int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1)) 
/(x**8 + 2*x**4 + 1),x)*x**4 - 88*int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x* 
*4 + 1))/(x**8 + 2*x**4 + 1),x))/(24*(x**4 + 1))