\(\int \sqrt {1-\frac {1}{1+2 x}} \, dx\) [30]

Optimal result

Integrand size = 15, antiderivative size = 37 \[ \int \sqrt {1-\frac {1}{1+2 x}} \, dx=\frac {\sqrt {x} \sqrt {1+2 x}}{\sqrt {2}}-\frac {1}{2} \text {arcsinh}\left (\sqrt {2} \sqrt {x}\right ) \] Output:

1/2*x^(1/2)*(1+2*x)^(1/2)*2^(1/2)-1/2*arcsinh(2^(1/2)*x^(1/2))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.95 \[ \int \sqrt {1-\frac {1}{1+2 x}} \, dx=\frac {\sqrt {\frac {x}{1+2 x}} \left (\sqrt {2} \sqrt {x} (1+2 x)+\sqrt {1+2 x} \log \left (-\sqrt {2} \sqrt {x}+\sqrt {1+2 x}\right )\right )}{2 \sqrt {x}} \] Input:

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

Output:

(Sqrt[x/(1 + 2*x)]*(Sqrt[2]*Sqrt[x]*(1 + 2*x) + Sqrt[1 + 2*x]*Log[-(Sqrt[2 
]*Sqrt[x]) + Sqrt[1 + 2*x]]))/(2*Sqrt[x])
 

Rubi [A] (warning: unable to verify)

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {239, 773, 51, 73, 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.

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

\(\Big \downarrow \) 239

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

\(\Big \downarrow \) 773

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 239

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Simp[1/Coefficient[v, x, 1 
]   Subst[Int[(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, n, p}, x] && Lin 
earQ[v, x] && NeQ[v, x]
 

rule 773

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 
2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] &&  !IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(26)=52\).

Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.73

method	result	size
trager	\(3 \left (\frac {1}{6}+\frac {x}{3}\right ) \sqrt {2}\, \sqrt {\frac {x}{1+2 x}}-\frac {\ln \left (4 x \sqrt {2}\, \sqrt {\frac {x}{1+2 x}}+2 \sqrt {2}\, \sqrt {\frac {x}{1+2 x}}+4 x +1\right )}{4}\)	\(64\)
default	\(\frac {\sqrt {2}\, \sqrt {\frac {x}{1+2 x}}\, \left (1+2 x \right ) \left (-\ln \left (\frac {\sqrt {2}}{4}+x \sqrt {2}+\sqrt {2 x^{2}+x}\right ) \sqrt {2}+4 \sqrt {2 x^{2}+x}\right )}{8 \sqrt {x \left (1+2 x \right )}}\)	\(69\)
risch	\(\frac {\left (1+2 x \right ) \sqrt {2}\, \sqrt {\frac {x}{1+2 x}}}{2}-\frac {\ln \left (\frac {\left (\frac {1}{2}+2 x \right ) \sqrt {2}}{2}+\sqrt {2 x^{2}+x}\right ) \sqrt {\frac {x}{1+2 x}}\, \sqrt {x \left (1+2 x \right )}}{4 x}\)	\(69\)

Input:

int((1-1/(1+2*x))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

3*(1/6+1/3*x)*2^(1/2)*(x/(1+2*x))^(1/2)-1/4*ln(4*x*2^(1/2)*(x/(1+2*x))^(1/ 
2)+2*2^(1/2)*(x/(1+2*x))^(1/2)+4*x+1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (26) = 52\).

Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.68 \[ \int \sqrt {1-\frac {1}{1+2 x}} \, dx=\frac {1}{2} \, \sqrt {2} {\left (2 \, x + 1\right )} \sqrt {\frac {x}{2 \, x + 1}} - \frac {1}{4} \, \log \left (\sqrt {2} \sqrt {\frac {x}{2 \, x + 1}} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {2} \sqrt {\frac {x}{2 \, x + 1}} - 1\right ) \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.83 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.43 \[ \int \sqrt {1-\frac {1}{1+2 x}} \, dx=\begin {cases} \sqrt {x} \sqrt {x + \frac {1}{2}} - \frac {\operatorname {acosh}{\left (\sqrt {2} \sqrt {x + \frac {1}{2}} \right )}}{2} & \text {for}\: \left |{x + \frac {1}{2}}\right | > \frac {1}{2} \\\frac {i \operatorname {asin}{\left (\sqrt {2} \sqrt {x + \frac {1}{2}} \right )}}{2} - \frac {i \left (x + \frac {1}{2}\right )^{\frac {3}{2}}}{\sqrt {- x}} + \frac {i \sqrt {x + \frac {1}{2}}}{2 \sqrt {- x}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(x)*sqrt(x + 1/2) - acosh(sqrt(2)*sqrt(x + 1/2))/2, Abs(x + 
 1/2) > 1/2), (I*asin(sqrt(2)*sqrt(x + 1/2))/2 - I*(x + 1/2)**(3/2)/sqrt(- 
x) + I*sqrt(x + 1/2)/(2*sqrt(-x)), True))
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).

Time = 0.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59 \[ \int \sqrt {1-\frac {1}{1+2 x}} \, dx=\frac {1}{8} \, \sqrt {2} {\left (\sqrt {2} \log \left ({\left | -2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} + x}\right )} - 1 \right |}\right ) \mathrm {sgn}\left (2 \, x + 1\right ) + 4 \, \sqrt {2 \, x^{2} + x} \mathrm {sgn}\left (2 \, x + 1\right )\right )} \] Input:

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

Output:

1/8*sqrt(2)*(sqrt(2)*log(abs(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 + x)) - 1) 
)*sgn(2*x + 1) + 4*sqrt(2*x^2 + x)*sgn(2*x + 1))
 

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \sqrt {1-\frac {1}{1+2 x}} \, dx=-\frac {\left (2\,x+1\right )\,\left (\frac {\ln \left (2\,x+\sqrt {{\left (2\,x+1\right )}^2-2\,x-1}+\frac {1}{2}\right )}{\sqrt {{\left (2\,x+1\right )}^2-2\,x-1}}-2\right )\,\sqrt {1-\frac {1}{2\,x+1}}}{4} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \sqrt {1-\frac {1}{1+2 x}} \, dx=\frac {\sqrt {x}\, \sqrt {2 x +1}\, \sqrt {2}}{2}-\frac {\mathrm {log}\left (\sqrt {2 x +1}+\sqrt {x}\, \sqrt {2}\right )}{2} \] Input:

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

Output:

(sqrt(x)*sqrt(2*x + 1)*sqrt(2) - log(sqrt(2*x + 1) + sqrt(x)*sqrt(2)))/2