3.31.99 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx\) [3099]

3.31.99.1 Optimal result

Integrand size = 25, antiderivative size = 553 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\frac {\sqrt {x^2+\sqrt {1+x^4}}}{a}+\frac {\left (1+a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {\left (-1-a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\frac {\left (\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}}-\sqrt {2} a^2 \sqrt {-a^2-\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2-\sqrt {1+a^4}}\right ) \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-a^2-\sqrt {1+a^4}} \left (-1+x^2+\sqrt {1+x^4}\right )}\right )}{2 a^2}+\frac {\left (-\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} a^2 \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2+\sqrt {1+a^4}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{a^2} \]

output

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

3.31.99.2 Mathematica [A] (verified)

Time = 4.63 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\frac {2 a \sqrt {x^2+\sqrt {1+x^4}}+\frac {2 \left (1+a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {2 \left (-1-a^4+\sqrt {1+a^4}\right ) \arctan \left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}} \left (1-a^2+\sqrt {1+a^4}\right ) \arctan \left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-a^2-\sqrt {1+a^4}} \left (-1+x^2+\sqrt {1+x^4}\right )}\right )+\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} \left (-1+a^2+\sqrt {1+a^4}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2} \]

input

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

output

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

3.31.99.3 Rubi [F]

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.

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.31.99.3.1 Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.31.99.4 Maple [F]

\[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{a x +1}d x\]

input

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

output

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

3.31.99.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\text {Timed out} \]

input

integrate((x^2+(x^4+1)^(1/2))^(1/2)/(a*x+1),x, algorithm="fricas")
 

output

Timed out
 

3.31.99.6 Sympy [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1}\, dx \]

input

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

output

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

3.31.99.7 Maxima [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1} \,d x } \]

input

integrate((x^2+(x^4+1)^(1/2))^(1/2)/(a*x+1),x, algorithm="maxima")
 

output

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

3.31.99.8 Giac [F]

\[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1} \,d x } \]

input

integrate((x^2+(x^4+1)^(1/2))^(1/2)/(a*x+1),x, algorithm="giac")
 

output

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

3.31.99.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx=\int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{a\,x+1} \,d x \]

input

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

output

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