\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(-1+x^2) \sqrt {1+x^4}} \, dx\) [1905]

Optimal result

Integrand size = 34, antiderivative size = 132 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \] Output:

1/2*(-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)))-1/2*(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 = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.22, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

int((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. 266 vs. \(2 (101) = 202\).

Time = 4.59 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \arctan \left (-\frac {{\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}}}{2 \, x}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} - x\right )} + 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (-\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} - x\right )} + 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} - 1}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx=\frac {\sqrt {2}\, \left (-\sqrt {\sqrt {2}-1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, \sqrt {2}+\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}-\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {2}\, x^{2}-\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{2}+\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}}{2 x}\right )-\sqrt {\sqrt {2}+1}\, \mathrm {log}\left (x^{2}-1\right )+\sqrt {\sqrt {2}+1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, \sqrt {2}\, x +2 \sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {x^{4}+1}\, x -\sqrt {x^{4}+1}\, \sqrt {2}+\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {2}\, x^{3}+\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, \sqrt {2}\, x -2 \sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}+x^{2}}\, x^{3}-2 x^{2}\right )\right )}{4} \] Input:

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

Output:

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