\(\int \frac {(-1+x^2) \sqrt {x^2+\sqrt {1+x^4}}}{(1+x^2) \sqrt {1+x^4}} \, dx\) [2302]

Optimal result

Integrand size = 39, antiderivative size = 176 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=-\sqrt {2 \left (-1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {2 \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \] 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 = 0.65 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=\sqrt {2} \left (-\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \left (-1+x^2+\sqrt {1+x^4}\right )}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {-1+x^2+\sqrt {1+x^4}}{\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}\right )\right ) \] 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[(Sqrt[1/2 + 1/Sqrt[2]]*(-1 + x^2 + Sq 
rt[1 + x^4]))/(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]]*ArcTan 
h[(-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 = 1.08 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.02, 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

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Sqrt[1 + I]*ArcTanh[(1 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/2 - (Sqrt[1 + 
 I]*ArcTanh[(1 + x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/2 + (Sqrt[1 - I]*ArcTa 
nh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/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 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. 319 vs. \(2 (138) = 276\).

Time = 3.03 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.81 \[ \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=\sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \arctan \left (-\frac {{\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}}}{2 \, x}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} + x\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} + x\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) \] 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)*sq 
rt(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 + sqr 
t(x^4 + 1))*sqrt(1/2*sqrt(2) + 1/2) + sqrt(x^4 + 1)*(sqrt(2) + 1))/(x^2 + 
1))
 

Sympy [F]

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

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