3.26.86 \(\int \frac {(-1+x^2)^2}{(1+x^2)^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx\) [2586]

3.26.86.1 Optimal result

Integrand size = 41, antiderivative size = 223 \[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x^2 \left (1+x^2\right )+x^2 \sqrt {1+x^4}}{x \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

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)+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)-(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)))
 

3.26.86.2 Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=\frac {x \left (1+x^2+\sqrt {1+x^4}\right )}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\sqrt {2} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \]

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

3.26.86.3 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

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.26.86.3.1 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] 
]
 

3.26.86.4 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)
 

3.26.86.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (179) = 358\).

Time = 1.20 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.43 \[ \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}}} \, dx=-\frac {2 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) + {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} {\left (x^{3} + 2 \, x\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )} + 3 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2} - 1} \log \left (-\frac {\sqrt {2} x^{2} - 2 \, x^{2} + \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )} \sqrt {-\sqrt {2} - 1} + {\left (x^{3} - \sqrt {2} {\left (x^{3} + 2 \, x\right )} + 3 \, x\right )} \sqrt {-\sqrt {2} - 1}\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) + {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2} - 1} \log \left (-\frac {\sqrt {2} x^{2} - 2 \, x^{2} - \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (\sqrt {x^{4} + 1} {\left (\sqrt {2} x - x\right )} \sqrt {-\sqrt {2} - 1} + {\left (x^{3} - \sqrt {2} {\left (x^{3} + 2 \, x\right )} + 3 \, x\right )} \sqrt {-\sqrt {2} - 1}\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )}}{x^{2} + 1}\right ) + 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x - x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, {\left (x^{2} + 1\right )}} \]

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)
 

3.26.86.6 Sympy [F]

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

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

3.26.86.7 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)
 

3.26.86.8 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)
 

3.26.86.9 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)