\(\int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx\) [96]

Optimal result

Integrand size = 34, antiderivative size = 232 \[ \int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {-x^2+\sqrt {1+x^4}}\right )+\frac {\arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {-x^2+\sqrt {1+x^4}}}{1-x^2+\sqrt {1+x^4}}\right )}{\sqrt {2 \left (-1+\sqrt {2}\right )}}-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {-x^2+\sqrt {1+x^4}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {-x^2+\sqrt {1+x^4}}}{1-x^2+\sqrt {1+x^4}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}} \] Output:

1/2*(2+2*2^(1/2))^(1/2)*arctan((2^(1/2)-1)^(1/2)*(-x^2+(x^4+1)^(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)))/(-2+2*2^(1/2))^(1/2)-1/2*(-2+2*2^(1/2))^(1/2)*arctanh((1+2^(1/2)) 
^(1/2)*(-x^2+(x^4+1)^(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)))/(2+2*2^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+\sqrt {2}} \left (\arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {-x^2+\sqrt {1+x^4}}\right )+\arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {-x^2+\sqrt {1+x^4}}}{1-x^2+\sqrt {1+x^4}}\right )\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {-x^2+\sqrt {1+x^4}}\right )+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {-x^2+\sqrt {1+x^4}}}{1-x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.33, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2558, 488, 219}

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.

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

\(\Big \downarrow \) 2558

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\text {arctanh}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )}{(1-i)^{3/2}}-\frac {\text {arctanh}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )}{(1+i)^{3/2}}\)

Input:

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

Output:

-(ArcTanh[(1 + I*x)/(Sqrt[1 - I]*Sqrt[1 - I*x^2])]/(1 - I)^(3/2)) - ArcTan 
h[(1 - I*x)/(Sqrt[1 + I]*Sqrt[1 + I*x^2])]/(1 + I)^(3/2)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 2558

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^ 
4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_Symbol] :> Simp[(1 - I)/2   Int[(c + d*x) 
^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Simp[(1 + I)/2   Int[(c + d*x)^m/Sqrt[ 
Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && G 
tQ[a, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 4.33 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (\frac {{\left (2 \, x^{2} + \sqrt {2} {\left (x^{3} - x^{2} + x + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2\right )} - 2 \, x\right )} \sqrt {-x^{2} + \sqrt {x^{4} + 1}} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}}}{x^{2} - 2 \, x + 1}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left (\frac {{\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {-x^{2} + \sqrt {x^{4} + 1}} + 2 \, {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left (\frac {{\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {-x^{2} + \sqrt {x^{4} + 1}} - 2 \, {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 2\right )} + 1\right )} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}}}{x^{2} + 2 \, x + 1}\right ) \] Input:

integrate((-x^2+(x^4+1)^(1/2))^(1/2)/(1+x)/(x^4+1)^(1/2),x, algorithm="fri 
cas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-x^2+(x^4+1)^(1/2))^(1/2)/(1+x)/(x^4+1)^(1/2),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

integrate((-x^2+(x^4+1)^(1/2))^(1/2)/(1+x)/(x^4+1)^(1/2),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {-x^2+\sqrt {1+x^4}}}{(1+x) \sqrt {1+x^4}} \, dx=\frac {\sqrt {2}\, \left (-\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {x^{4}+1}\, \sqrt {2}\, x -\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {x^{4}+1}\, x -\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {x^{4}+1}+\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {2}\, x^{3}+\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {2}-\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, x^{3}-\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, x^{2}+\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, x -\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}}{2 x^{2}-2}\right )-\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (x^{2}+2 x +1\right )+\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {x^{4}+1}\, \sqrt {2}\, x +\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {x^{4}+1}\, x -\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {x^{4}+1}+\sqrt {x^{4}+1}\, \sqrt {2}+\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {2}\, x^{3}-\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, \sqrt {2}+\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, x^{3}-\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, x^{2}-\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}\, x -\sqrt {\sqrt {2}-1}\, \sqrt {\sqrt {x^{4}+1}-x^{2}}+x^{2}+1\right )\right )}{4} \] Input:

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