\(\int \frac {(-1+x^2)^2 \sqrt {x^2+\sqrt {1+x^4}}}{(1+x^2)^2 \sqrt {1+x^4}} \, dx\) [2590]

Optimal result

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

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx=-\frac {x \left (-1+x^2+\sqrt {1+x^4}\right )}{\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \] Input:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.49 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {7293, 2009}

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \sqrt {1+i} \text {arctanh}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{(1+i)^{5/2}}-\frac {1}{2} \sqrt {1+i} \text {arctanh}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {\text {arctanh}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{(1+i)^{5/2}}+\frac {1}{2} \sqrt {1-i} \text {arctanh}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {\text {arctanh}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{(1-i)^{5/2}}-\frac {1}{2} \sqrt {1-i} \text {arctanh}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {\text {arctanh}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{(1-i)^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}}+\frac {i \sqrt {1-i x^2}}{2 (-x+i)}-\frac {i \sqrt {1-i x^2}}{2 (x+i)}-\frac {i \sqrt {1+i x^2}}{2 (-x+i)}+\frac {i \sqrt {1+i x^2}}{2 (x+i)}\)

Input:

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

Output:

((I/2)*Sqrt[1 - I*x^2])/(I - x) - ((I/2)*Sqrt[1 - I*x^2])/(I + x) - ((I/2) 
*Sqrt[1 + I*x^2])/(I - x) + ((I/2)*Sqrt[1 + I*x^2])/(I + x) + ArcTanh[(1 - 
 x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])]/(1 + I)^(5/2) + (Sqrt[1 + I]*ArcTanh[(1 
 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/2 - ArcTanh[(1 + x)/(Sqrt[1 + I]*Sqr 
t[1 - I*x^2])]/(1 + I)^(5/2) - (Sqrt[1 + I]*ArcTanh[(1 + x)/(Sqrt[1 + I]*S 
qrt[1 - I*x^2])])/2 + ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])]/(1 - 
I)^(5/2) + (Sqrt[1 - I]*ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/2 
- ArcTanh[(1 + x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])]/(1 - I)^(5/2) - (Sqrt[1 - 
 I]*ArcTanh[(1 + x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/2 + ArcTanh[(Sqrt[2]*x 
)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]
 

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] 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx=-\frac {2 \, {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} + x^{2} - \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )} + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1}}{2 \, x}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) + 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, {\left (x^{2} + 1\right )}} \] Input:

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

Output:

-1/4*(2*(x^2 + 1)*sqrt(sqrt(2) - 1)*arctan(-1/2*(sqrt(2)*x^2 + x^2 - sqrt( 
x^4 + 1)*(sqrt(2) + 1) + 1)*sqrt(x^2 + sqrt(x^4 + 1))*sqrt(sqrt(2) - 1)/x) 
 - sqrt(2)*(x^2 + 1)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 + 2*(sqrt(2)*x^3 + sq 
rt(2)*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1) + (x^2 + 1)*sqrt(sqr 
t(2) + 1)*log((sqrt(2)*x^2 + 2*x^2 + (x^3 + sqrt(2)*x - sqrt(x^4 + 1)*x + 
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 - sqrt(x^4 + 1)*x + x)*sqrt(x^2 + sqrt(x^4 + 1))*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)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx=-\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-\sqrt {2}\, x \right )}{4}+\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+\sqrt {2}\, x \right )}{4}-4 \left (\int \frac {\sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x^{2}}{x^{8}+2 x^{6}+2 x^{4}+2 x^{2}+1}d x \right ) \] Input:

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

Output:

( - sqrt(2)*log(sqrt(sqrt(x**4 + 1) + x**2) - sqrt(2)*x) + sqrt(2)*log(sqr 
t(sqrt(x**4 + 1) + x**2) + sqrt(2)*x) - 16*int((sqrt(sqrt(x**4 + 1) + x**2 
)*sqrt(x**4 + 1)*x**2)/(x**8 + 2*x**6 + 2*x**4 + 2*x**2 + 1),x))/4