\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(-1+x^4) \sqrt {1+x^4}} \, dx\) [2804]

Optimal result

Integrand size = 34, antiderivative size = 273 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \] Output:

1/4*(-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)))-1/4*(-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)))-1/4*(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)))-1/4*(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 = 1.07 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=\frac {\sqrt {-1+\sqrt {2}} \left (-\arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{2 \sqrt {2}} \] Input:

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

Output:

(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]])] + ArcTan[(-1 + x^2 + Sqrt[1 + x^4])/( 
Sqrt[2*(1 + Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 + x^4]])]) - Sqrt[1 + Sqrt[2]]*A 
rcTanh[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sqrt[1 + x^4]))/(x*Sqrt[x^2 + Sq 
rt[1 + x^4]])] - Sqrt[1 + Sqrt[2]]*ArcTanh[(-1 + x^2 + Sqrt[1 + x^4])/(Sqr 
t[2*(1 + Sqrt[2])]*x*Sqrt[x^2 + Sqrt[1 + x^4]])])/(2*Sqrt[2])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.68 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1+i} \text {arctanh}\left (\frac {1+i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \text {arctanh}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )\)

Input:

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

Output:

(Sqrt[1 + I]*ArcTanh[(1 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/8 - (Sqrt[1 - 
 I]*ArcTanh[(1 - I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])])/8 + (Sqrt[1 - I]*Arc 
Tanh[(1 + I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])])/8 - (Sqrt[1 + I]*ArcTanh[(1 
 + x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/8 + (Sqrt[1 - I]*ArcTanh[(1 - x)/(Sq 
rt[1 - I]*Sqrt[1 + I*x^2])])/8 + (Sqrt[1 + I]*ArcTanh[(1 - I*x)/(Sqrt[1 + 
I]*Sqrt[1 + I*x^2])])/8 - (Sqrt[1 + I]*ArcTanh[(1 + I*x)/(Sqrt[1 + I]*Sqrt 
[1 + I*x^2])])/8 - (Sqrt[1 - I]*ArcTanh[(1 + x)/(Sqrt[1 - I]*Sqrt[1 + I*x^ 
2])])/8
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 8.42 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^4\right ) \sqrt {1+x^4}} \, dx=-\frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \arctan \left (-\frac {2 \, {\left (6 \, x^{7} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{7} - x^{3}\right )} - {\left (8 \, x^{5} + \sqrt {2} {\left (5 \, x^{5} - x\right )}\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}}}{7 \, x^{8} + 10 \, x^{4} - 1}\right ) - \frac {1}{8} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (4 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + 2 \, x^{2}\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}}}{x^{4} - 1}\right ) + \frac {1}{8} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (\frac {2 \, {\left (\sqrt {2} x^{3} + 2 \, x^{3} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x + x\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (4 \, x^{4} + \sqrt {2} {\left (3 \, x^{4} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} x^{2} + 2 \, x^{2}\right )}\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}}}{x^{4} - 1}\right ) \] Input:

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

Output:

-1/4*sqrt(1/2*sqrt(2) - 1/2)*arctan(-2*(6*x^7 + 2*x^3 + 2*sqrt(2)*(x^7 - x 
^3) - (8*x^5 + sqrt(2)*(5*x^5 - x))*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1 
))*sqrt(1/2*sqrt(2) - 1/2)/(7*x^8 + 10*x^4 - 1)) - 1/8*sqrt(1/2*sqrt(2) + 
1/2)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x + x))*sqrt(x^2 
 + sqrt(x^4 + 1)) + (4*x^4 + sqrt(2)*(3*x^4 + 1) + 2*sqrt(x^4 + 1)*(sqrt(2 
)*x^2 + 2*x^2))*sqrt(1/2*sqrt(2) + 1/2))/(x^4 - 1)) + 1/8*sqrt(1/2*sqrt(2) 
 + 1/2)*log((2*(sqrt(2)*x^3 + 2*x^3 + sqrt(x^4 + 1)*(sqrt(2)*x + x))*sqrt( 
x^2 + sqrt(x^4 + 1)) - (4*x^4 + sqrt(2)*(3*x^4 + 1) + 2*sqrt(x^4 + 1)*(sqr 
t(2)*x^2 + 2*x^2))*sqrt(1/2*sqrt(2) + 1/2))/(x^4 - 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(sqrt(x**4 + 1) + x**2)*sqrt(x**4 + 1))/(x**8 - 1),x)