\(\int \frac {1+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx\) [2132]

Optimal result

Integrand size = 31, antiderivative size = 154 \[ \int \frac {1+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {104-36 x+214 x^2-24 x^3+\left (40+48 x+72 x^2+16 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-24+214 x-24 x^2+\left (40+72 x+16 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )}{48 x \sqrt {1+x^2}+24 \left (1+2 x^2\right )}+\frac {5}{4} \log \left (x+\sqrt {1+x^2}\right )-4 \log \left (1+\sqrt {x+\sqrt {1+x^2}}\right ) \] Output:

(104-36*x+214*x^2-24*x^3+(16*x^3+72*x^2+48*x+40)*(x+(x^2+1)^(1/2))^(1/2)+( 
x^2+1)^(1/2)*(-24+214*x-24*x^2+(16*x^2+72*x+40)*(x+(x^2+1)^(1/2))^(1/2)))/ 
(48*x*(x^2+1)^(1/2)+48*x^2+24)+5/4*ln(x+(x^2+1)^(1/2))-4*ln(1+(x+(x^2+1)^( 
1/2))^(1/2))
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \frac {1+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {104-36 x+214 x^2-24 x^3+\left (40+48 x+72 x^2+16 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-24+214 x-24 x^2+\left (40+72 x+16 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )}{48 x \sqrt {1+x^2}+24 \left (1+2 x^2\right )}+\frac {5}{4} \log \left (x+\sqrt {1+x^2}\right )-4 \log \left (1+\sqrt {x+\sqrt {1+x^2}}\right ) \] Input:

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

Output:

(104 - 36*x + 214*x^2 - 24*x^3 + (40 + 48*x + 72*x^2 + 16*x^3)*Sqrt[x + Sq 
rt[1 + x^2]] + Sqrt[1 + x^2]*(-24 + 214*x - 24*x^2 + (40 + 72*x + 16*x^2)* 
Sqrt[x + Sqrt[1 + x^2]]))/(48*x*Sqrt[1 + x^2] + 24*(1 + 2*x^2)) + (5*Log[x 
 + Sqrt[1 + x^2]])/4 - 4*Log[1 + Sqrt[x + Sqrt[1 + x^2]]]
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arcsinh}(x)}{4}+\frac {1}{2} \text {arctanh}\left (\sqrt {x^2+1}\right )-2 \text {arctanh}\left (\sqrt {\sqrt {x^2+1}+x}\right )-\frac {x^2}{4}+\frac {1}{4} \sqrt {x^2+1} x+\frac {1}{6} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {\sqrt {x^2+1}}{2}+\frac {3}{2} \sqrt {\sqrt {x^2+1}+x}+\frac {3}{2 \sqrt {\sqrt {x^2+1}+x}}-\frac {1}{2 \left (\sqrt {x^2+1}+x\right )}+\frac {1}{6 \left (\sqrt {x^2+1}+x\right )^{3/2}}+\frac {1}{2} \log \left (\sqrt {x^2+1}+x\right )-2 \log \left (\sqrt {\sqrt {x^2+1}+x}+1\right )-\frac {\log (x)}{2}\)

Input:

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

Output:

-1/4*x^2 - Sqrt[1 + x^2]/2 + (x*Sqrt[1 + x^2])/4 + 1/(6*(x + Sqrt[1 + x^2] 
)^(3/2)) - 1/(2*(x + Sqrt[1 + x^2])) + 3/(2*Sqrt[x + Sqrt[1 + x^2]]) + (3* 
Sqrt[x + Sqrt[1 + x^2]])/2 + (x + Sqrt[1 + x^2])^(3/2)/6 + ArcSinh[x]/4 + 
ArcTanh[Sqrt[1 + x^2]]/2 - 2*ArcTanh[Sqrt[x + Sqrt[1 + x^2]]] - Log[x]/2 + 
 Log[x + Sqrt[1 + x^2]]/2 - 2*Log[1 + Sqrt[x + Sqrt[1 + x^2]]]
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.55 \[ \int \frac {1+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{4} \, x^{2} + \frac {1}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} {\left (x - 5\right )} - 4 \, x + 5\right )} \sqrt {x + \sqrt {x^{2} + 1}} + \frac {1}{4} \, \sqrt {x^{2} + 1} {\left (x - 4\right )} + \frac {1}{2} \, x - 4 \, \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) + \frac {5}{2} \, \log \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) \] Input:

integrate((1+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="fric 
as")
 

Output:

-1/4*x^2 + 1/3*(x^2 - sqrt(x^2 + 1)*(x - 5) - 4*x + 5)*sqrt(x + sqrt(x^2 + 
 1)) + 1/4*sqrt(x^2 + 1)*(x - 4) + 1/2*x - 4*log(sqrt(x + sqrt(x^2 + 1)) + 
 1) + 5/2*log(sqrt(x + sqrt(x^2 + 1)))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((1+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

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

Output:

integrate((sqrt(x^2 + 1) + 1)/(sqrt(x + sqrt(x^2 + 1)) + 1), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {\sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}\, x}{4}+\sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}+\frac {\sqrt {x^{2}+1}\, x}{4}-\sqrt {x^{2}+1}-\frac {\sqrt {\sqrt {x^{2}+1}+x}}{2}-\left (\int \frac {\sqrt {\sqrt {x^{2}+1}+x}}{x^{2}+1}d x \right )+\frac {5 \left (\int \frac {\sqrt {\sqrt {x^{2}+1}+x}\, x^{3}}{x^{2}+1}d x \right )}{8}-\left (\int \frac {\sqrt {\sqrt {x^{2}+1}+x}\, x^{2}}{x^{2}+1}d x \right )+\int \frac {\sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{x^{2}+1}d x +\frac {5 \left (\int \frac {\sqrt {\sqrt {x^{2}+1}+x}\, x}{x^{2}+1}d x \right )}{8}-\mathrm {log}\left (\sqrt {x^{2}+1}+x -1\right )+\mathrm {log}\left (\sqrt {x^{2}+1}+x +1\right )+\frac {\mathrm {log}\left (\sqrt {x^{2}+1}+x \right )}{4}+2 \,\mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}-1\right )-2 \,\mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}+1\right )-\mathrm {log}\left (x \right )-\frac {x^{2}}{4}+\frac {x}{2} \] Input:

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

Output:

( - 2*sqrt(sqrt(x**2 + 1) + x)*sqrt(x**2 + 1)*x + 8*sqrt(sqrt(x**2 + 1) + 
x)*sqrt(x**2 + 1) + 2*sqrt(x**2 + 1)*x - 8*sqrt(x**2 + 1) - 4*sqrt(sqrt(x* 
*2 + 1) + x) - 8*int(sqrt(sqrt(x**2 + 1) + x)/(x**2 + 1),x) + 5*int((sqrt( 
sqrt(x**2 + 1) + x)*x**3)/(x**2 + 1),x) - 8*int((sqrt(sqrt(x**2 + 1) + x)* 
x**2)/(x**2 + 1),x) + 8*int((sqrt(sqrt(x**2 + 1) + x)*sqrt(x**2 + 1))/(x** 
2 + 1),x) + 5*int((sqrt(sqrt(x**2 + 1) + x)*x)/(x**2 + 1),x) - 8*log(sqrt( 
x**2 + 1) + x - 1) + 8*log(sqrt(x**2 + 1) + x + 1) + 2*log(sqrt(x**2 + 1) 
+ x) + 16*log(sqrt(sqrt(x**2 + 1) + x) - 1) - 16*log(sqrt(sqrt(x**2 + 1) + 
 x) + 1) - 8*log(x) - 2*x**2 + 4*x)/8