\(\int \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} \, dx\) [18]

Optimal result

Integrand size = 17, antiderivative size = 96 \[ \int \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} \, dx=\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} x+\frac {1}{4} \arctan \left (\frac {3+\sqrt {1+\frac {1}{x}}}{2 \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}\right )-\frac {3}{4} \text {arctanh}\left (\frac {1-3 \sqrt {1+\frac {1}{x}}}{2 \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}\right ) \] Output:

1/4*arctan(1/2*(3+(1+1/x)^(1/2))/(1/x+(1+1/x)^(1/2))^(1/2))-3/4*arctanh(1/ 
2*(1-3*(1+1/x)^(1/2))/(1/x+(1+1/x)^(1/2))^(1/2))+x*(1/x+(1+1/x)^(1/2))^(1/ 
2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} \, dx=\frac {1}{2} \left (2 \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} x+\arctan \left (1+\sqrt {1+\frac {1}{x}}-\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}\right )+3 \text {arctanh}\left (1-\sqrt {1+\frac {1}{x}}+\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}\right )\right ) \] Input:

Integrate[Sqrt[Sqrt[1 + x^(-1)] + x^(-1)],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {7268, 1347, 27, 1366, 25, 1154, 217, 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.

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

\(\Big \downarrow \) 7268

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

\(\Big \downarrow \) 1347

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1366

\(\displaystyle -2 \left (\frac {1}{4} \left (-\frac {3}{2} \int \frac {1}{\left (1-\sqrt {1+\frac {1}{x}}\right ) \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}d\sqrt {1+\frac {1}{x}}-\frac {1}{2} \int -\frac {1}{\left (\sqrt {1+\frac {1}{x}}+1\right ) \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}d\sqrt {1+\frac {1}{x}}\right )-\frac {1}{2} \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{\left (\sqrt {1+\frac {1}{x}}+1\right ) \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}d\sqrt {1+\frac {1}{x}}-\frac {3}{2} \int \frac {1}{\left (1-\sqrt {1+\frac {1}{x}}\right ) \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}d\sqrt {1+\frac {1}{x}}\right )-\frac {1}{2} \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle -2 \left (\frac {1}{4} \left (3 \int \frac {1}{3-\frac {1}{x}}d\frac {1-3 \sqrt {1+\frac {1}{x}}}{\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}-\int \frac {1}{-5-\frac {1}{x}}d\left (-\frac {\sqrt {1+\frac {1}{x}}+3}{\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}\right )\right )-\frac {1}{2} \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x\right )\)

\(\Big \downarrow \) 217

\(\displaystyle -2 \left (\frac {1}{4} \left (3 \int \frac {1}{3-\frac {1}{x}}d\frac {1-3 \sqrt {1+\frac {1}{x}}}{\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1}{x}+1}+3}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right )\right )-\frac {1}{2} \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -2 \left (\frac {1}{4} \left (\frac {3}{2} \text {arctanh}\left (\frac {1-3 \sqrt {\frac {1}{x}+1}}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt {\frac {1}{x}+1}+3}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right )\right )-\frac {1}{2} \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x\right )\)

Input:

Int[Sqrt[Sqrt[1 + x^(-1)] + x^(-1)],x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1347

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f 
_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a*h - g*c*x)*(a + c*x^2)^(p + 1)*((d + 
 e*x + f*x^2)^q/(2*a*c*(p + 1))), x] + Simp[2/(4*a*c*(p + 1))   Int[(a + c* 
x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[g*c*d*(2*p + 3) - a*(h*e*q) + ( 
g*c*e*(2*p + q + 3) - a*(2*h*f*q))*x + g*c*f*(2*p + 2*q + 3)*x^2, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] & 
& GtQ[q, 0]
 

rule 1366

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( 
f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q 
)))   Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(h/2 - c*(g/( 
2*q)))   Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d 
, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
 

rule 7268

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears 
[u, x]}, Simp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls 
t[[2]])], x] /;  !FalseQ[lst]]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.68 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.27 \[ \int \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} \, dx=x \sqrt {\frac {x \sqrt {\frac {x + 1}{x}} + 1}{x}} + \frac {1}{4} \, \arctan \left (\frac {2 \, {\left (x \sqrt {\frac {x + 1}{x}} - 3 \, x\right )} \sqrt {\frac {x \sqrt {\frac {x + 1}{x}} + 1}{x}}}{8 \, x - 1}\right ) + \frac {3}{4} \, \log \left (2 \, {\left (x \sqrt {\frac {x + 1}{x}} + x\right )} \sqrt {\frac {x \sqrt {\frac {x + 1}{x}} + 1}{x}} + 2 \, x \sqrt {\frac {x + 1}{x}} + 2 \, x + 3\right ) \] Input:

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

Output:

x*sqrt((x*sqrt((x + 1)/x) + 1)/x) + 1/4*arctan(2*(x*sqrt((x + 1)/x) - 3*x) 
*sqrt((x*sqrt((x + 1)/x) + 1)/x)/(8*x - 1)) + 3/4*log(2*(x*sqrt((x + 1)/x) 
 + x)*sqrt((x*sqrt((x + 1)/x) + 1)/x) + 2*x*sqrt((x + 1)/x) + 2*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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