\(\int \frac {(1+x^2) \sqrt {x^2+\sqrt {1+x^4}}}{(-1+x^2) \sqrt {1+x^4}} \, dx\) [2298]

Optimal result

Integrand size = 39, antiderivative size = 175 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\sqrt {2 \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {2 \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \] Output:

(-2+2*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)))+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)-(2+2*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 = 0.59 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\sqrt {2} \left (\sqrt {-1+\sqrt {2}} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {7276, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(Sqrt[1 - I]*ArcTanh[(1 - I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])]) + (Sqr 
t[1 - I]*ArcTanh[(1 + I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])])/2 + (Sqrt[1 + I 
]*ArcTanh[(1 - I*x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])])/2 - (Sqrt[1 + I]*ArcTa 
nh[(1 + I*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 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (137) = 274\).

Time = 2.84 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.86 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=-\sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \arctan \left (-\frac {{\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}}}{2 \, x}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} - x\right )} + 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} - x\right )} + 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.81 \[ \int \frac {\left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\frac {\sqrt {2}\, \left (-2 \sqrt {\sqrt {2}-1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, \sqrt {2}+\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}-\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {2}\, x^{2}-\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}+\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}}{2 x}\right )-2 \sqrt {\sqrt {2}+1}\, \mathrm {log}\left (x^{2}-1\right )+2 \sqrt {\sqrt {2}+1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, \sqrt {2}\, x +2 \sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x -\sqrt {x^{4}+1}\, \sqrt {2}+\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {2}\, x^{3}+\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {2}\, x -2 \sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{3}-2 x^{2}\right )-\mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}-\sqrt {2}\, x \right )+\mathrm {log}\left (\sqrt {\sqrt {x^{4}+1}+x^{2}}+\sqrt {2}\, x \right )\right )}{4} \] Input:

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

Output:

(sqrt(2)*( - 2*sqrt(sqrt(2) - 1)*atan((sqrt(sqrt(2) - 1)*sqrt(sqrt(x**4 + 
1) + x**2)*sqrt(x**4 + 1)*sqrt(2) + sqrt(sqrt(2) - 1)*sqrt(sqrt(x**4 + 1) 
+ x**2)*sqrt(x**4 + 1) - sqrt(sqrt(2) - 1)*sqrt(sqrt(x**4 + 1) + x**2)*sqr 
t(2)*x**2 - sqrt(sqrt(2) - 1)*sqrt(sqrt(x**4 + 1) + x**2)*x**2 + sqrt(sqrt 
(2) - 1)*sqrt(sqrt(x**4 + 1) + x**2))/(2*x)) - 2*sqrt(sqrt(2) + 1)*log(x** 
2 - 1) + 2*sqrt(sqrt(2) + 1)*log( - sqrt(sqrt(2) + 1)*sqrt(sqrt(x**4 + 1) 
+ x**2)*sqrt(x**4 + 1)*sqrt(2)*x + 2*sqrt(sqrt(2) + 1)*sqrt(sqrt(x**4 + 1) 
 + x**2)*sqrt(x**4 + 1)*x - sqrt(x**4 + 1)*sqrt(2) + sqrt(sqrt(2) + 1)*sqr 
t(sqrt(x**4 + 1) + x**2)*sqrt(2)*x**3 + sqrt(sqrt(2) + 1)*sqrt(sqrt(x**4 + 
 1) + x**2)*sqrt(2)*x - 2*sqrt(sqrt(2) + 1)*sqrt(sqrt(x**4 + 1) + x**2)*x* 
*3 - 2*x**2) - log(sqrt(sqrt(x**4 + 1) + x**2) - sqrt(2)*x) + log(sqrt(sqr 
t(x**4 + 1) + x**2) + sqrt(2)*x)))/4