\(\int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [2792]

Optimal result

Integrand size = 29, antiderivative size = 269 \[ \int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {(6 b-16 a x) \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+(-9 b+8 a x) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (-16 \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+8 \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{12 a^2 x+12 a \sqrt {-b+a^2 x^2}}+\frac {3 b \text {arctanh}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{4 a} \] Output:

((-16*a*x+6*b)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(8*a*x-9*b)*(a*x+(a 
^2*x^2-b)^(1/2))^(1/2)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(a^2*x^2-b) 
^(1/2)*(-16*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+8*(a*x+(a^2*x^2-b)^(1/ 
2))^(1/2)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)))/(12*a^2*x+12*a*(a^2*x^ 
2-b)^(1/2))+3/4*b*arctanh((1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2))/a
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}} \left (b \left (6-9 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+8 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-2+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\right )}{a x+\sqrt {-b+a^2 x^2}}+9 b \text {arctanh}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{12 a} \] Input:

Integrate[1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]
 

Output:

((Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]*(b*(6 - 9*Sqrt[a*x + Sqrt[-b + 
a^2*x^2]]) + 8*(a*x + Sqrt[-b + a^2*x^2])*(-2 + Sqrt[a*x + Sqrt[-b + a^2*x 
^2]])))/(a*x + Sqrt[-b + a^2*x^2]) + 9*b*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[ 
-b + a^2*x^2]]]])/(12*a)
 

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.

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

\(\Big \downarrow \) 7299

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

Input:

Int[1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]
 

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} + 1\right ) - 9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} - 1\right ) + 2 \, {\left (6 \, a x - {\left (9 \, a x - 9 \, \sqrt {a^{2} x^{2} - b} - 8\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 6 \, \sqrt {a^{2} x^{2} - b} - 16\right )} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}{24 \, a} \] Input:

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

Output:

1/24*(9*b*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) + 1) - 9*b*log(sqrt( 
sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) - 1) + 2*(6*a*x - (9*a*x - 9*sqrt(a^2*x 
^2 - b) - 8)*sqrt(a*x + sqrt(a^2*x^2 - b)) - 6*sqrt(a^2*x^2 - b) - 16)*sqr 
t(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1))/a
 

Sympy [F]

\[ \int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\, dx \] Input:

integrate(1/(1+(a*x+(a**2*x**2-b)**(1/2))**(1/2))**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {1}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}} \,d x } \] Input:

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

 - int(sqrt(sqrt(sqrt(a**2*x**2 - b) + a*x) + 1)/(sqrt(a**2*x**2 - b) + a* 
x - 1),x) + int((sqrt(sqrt(a**2*x**2 - b) + a*x)*sqrt(sqrt(sqrt(a**2*x**2 
- b) + a*x) + 1))/(sqrt(a**2*x**2 - b) + a*x - 1),x)