3.3.66 \(\int \frac {\log (1+\sqrt {x}-x)}{x} \, dx\) [266]

3.3.66.1 Optimal result

Integrand size = 15, antiderivative size = 122 \[ \int \frac {\log \left (1+\sqrt {x}-x\right )}{x} \, dx=-2 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (1+\sqrt {5}-2 \sqrt {x}\right )-2 \log \left (1-\frac {2 \sqrt {x}}{1-\sqrt {5}}\right ) \log \left (\sqrt {x}\right )+2 \log \left (1+\sqrt {x}-x\right ) \log \left (\sqrt {x}\right )+2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {x}}{1+\sqrt {5}}\right )-2 \operatorname {PolyLog}\left (2,\frac {2 \sqrt {x}}{1-\sqrt {5}}\right ) \]

output

-2*ln(1/2+1/2*5^(1/2))*ln(1+5^(1/2)-2*x^(1/2))+ln(x)*ln(1-x+x^(1/2))-ln(x) 
*ln(1-2*x^(1/2)/(-5^(1/2)+1))-2*polylog(2,2*x^(1/2)/(-5^(1/2)+1))+2*polylo 
g(2,1-2*x^(1/2)/(5^(1/2)+1))
 

3.3.66.2 Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99 \[ \int \frac {\log \left (1+\sqrt {x}-x\right )}{x} \, dx=-2 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (1+\sqrt {5}-2 \sqrt {x}\right )+\left (\log \left (-1+\sqrt {5}\right )-\log \left (-1+\sqrt {5}+2 \sqrt {x}\right )\right ) \log (x)+\log \left (1+\sqrt {x}-x\right ) \log (x)+2 \operatorname {PolyLog}\left (2,\frac {1+\sqrt {5}-2 \sqrt {x}}{1+\sqrt {5}}\right )-2 \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x}}{-1+\sqrt {5}}\right ) \]

input

Integrate[Log[1 + Sqrt[x] - x]/x,x]
 

output

-2*Log[(1 + Sqrt[5])/2]*Log[1 + Sqrt[5] - 2*Sqrt[x]] + (Log[-1 + Sqrt[5]] 
- Log[-1 + Sqrt[5] + 2*Sqrt[x]])*Log[x] + Log[1 + Sqrt[x] - x]*Log[x] + 2* 
PolyLog[2, (1 + Sqrt[5] - 2*Sqrt[x])/(1 + Sqrt[5])] - 2*PolyLog[2, (-2*Sqr 
t[x])/(-1 + Sqrt[5])]
 

3.3.66.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3010, 3004, 2804, 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.

\(\displaystyle \int \frac {\log \left (-x+\sqrt {x}+1\right )}{x} \, dx\)

\(\Big \downarrow \) 3010

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

\(\Big \downarrow \) 3004

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

\(\Big \downarrow \) 2804

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {x}}{1+\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x}}{1-\sqrt {5}}\right )-\log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (-2 \sqrt {x}+\sqrt {5}+1\right )-\log \left (1-\frac {2 \sqrt {x}}{1-\sqrt {5}}\right ) \log \left (\sqrt {x}\right )+\log \left (-x+\sqrt {x}+1\right ) \log \left (\sqrt {x}\right )\right )\)

input

Int[Log[1 + Sqrt[x] - x]/x,x]
 

output

2*(-(Log[(1 + Sqrt[5])/2]*Log[1 + Sqrt[5] - 2*Sqrt[x]]) - Log[1 - (2*Sqrt[ 
x])/(1 - Sqrt[5])]*Log[Sqrt[x]] + Log[1 + Sqrt[x] - x]*Log[Sqrt[x]] + Poly 
Log[2, 1 - (2*Sqrt[x])/(1 + Sqrt[5])] - PolyLog[2, (2*Sqrt[x])/(1 - Sqrt[5 
])])
 

3.3.66.3.1 Defintions of rubi rules used

rule 2009

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

rule 2804

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ 
u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / 
; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
 

rule 3004

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[Log[d + e*x]*((a + b*Log[c*RFx^p])^n/e), x] - Simp[b*n*(p/e) 
   Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x] /; 
 FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]
 

rule 3010

Int[((a_.) + Log[u_]*(b_.))*(RFx_), x_Symbol] :> With[{lst = SubstForFracti 
onalPowerOfLinear[RFx*(a + b*Log[u]), x]}, Simp[lst[[2]]*lst[[4]]   Subst[I 
nt[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x] /;  !FalseQ[lst]] /; FreeQ[{ 
a, b}, x] && RationalFunctionQ[RFx, x]
 

3.3.66.4 Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\ln \left (x \right ) \ln \left (1-x +\sqrt {x}\right )-\ln \left (x \right ) \ln \left (\frac {1+\sqrt {5}-2 \sqrt {x}}{\sqrt {5}+1}\right )-\ln \left (x \right ) \ln \left (\frac {-1+\sqrt {5}+2 \sqrt {x}}{\sqrt {5}-1}\right )-2 \operatorname {dilog}\left (\frac {1+\sqrt {5}-2 \sqrt {x}}{\sqrt {5}+1}\right )-2 \operatorname {dilog}\left (\frac {-1+\sqrt {5}+2 \sqrt {x}}{\sqrt {5}-1}\right )\)	\(102\)
default	\(\ln \left (x \right ) \ln \left (1-x +\sqrt {x}\right )-\ln \left (x \right ) \ln \left (\frac {1+\sqrt {5}-2 \sqrt {x}}{\sqrt {5}+1}\right )-\ln \left (x \right ) \ln \left (\frac {-1+\sqrt {5}+2 \sqrt {x}}{\sqrt {5}-1}\right )-2 \operatorname {dilog}\left (\frac {1+\sqrt {5}-2 \sqrt {x}}{\sqrt {5}+1}\right )-2 \operatorname {dilog}\left (\frac {-1+\sqrt {5}+2 \sqrt {x}}{\sqrt {5}-1}\right )\)	\(102\)
parts	\(\ln \left (x \right ) \ln \left (1-x +\sqrt {x}\right )-\ln \left (x \right ) \ln \left (\frac {1+\sqrt {5}-2 \sqrt {x}}{\sqrt {5}+1}\right )-\ln \left (x \right ) \ln \left (\frac {-1+\sqrt {5}+2 \sqrt {x}}{\sqrt {5}-1}\right )-2 \operatorname {dilog}\left (\frac {1+\sqrt {5}-2 \sqrt {x}}{\sqrt {5}+1}\right )-2 \operatorname {dilog}\left (\frac {-1+\sqrt {5}+2 \sqrt {x}}{\sqrt {5}-1}\right )\)	\(102\)

input

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

output

ln(x)*ln(1-x+x^(1/2))-ln(x)*ln((1+5^(1/2)-2*x^(1/2))/(5^(1/2)+1))-ln(x)*ln 
((-1+5^(1/2)+2*x^(1/2))/(5^(1/2)-1))-2*dilog((1+5^(1/2)-2*x^(1/2))/(5^(1/2 
)+1))-2*dilog((-1+5^(1/2)+2*x^(1/2))/(5^(1/2)-1))
 

3.3.66.5 Fricas [F]

\[ \int \frac {\log \left (1+\sqrt {x}-x\right )}{x} \, dx=\int { \frac {\log \left (-x + \sqrt {x} + 1\right )}{x} \,d x } \]

input

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

output

integral(log(-x + sqrt(x) + 1)/x, x)
 

3.3.66.6 Sympy [F]

\[ \int \frac {\log \left (1+\sqrt {x}-x\right )}{x} \, dx=\int \frac {\log {\left (\sqrt {x} - x + 1 \right )}}{x}\, dx \]

input

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

output

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

3.3.66.7 Maxima [F]

\[ \int \frac {\log \left (1+\sqrt {x}-x\right )}{x} \, dx=\int { \frac {\log \left (-x + \sqrt {x} + 1\right )}{x} \,d x } \]

input

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

output

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

3.3.66.8 Giac [F]

\[ \int \frac {\log \left (1+\sqrt {x}-x\right )}{x} \, dx=\int { \frac {\log \left (-x + \sqrt {x} + 1\right )}{x} \,d x } \]

input

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

output

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

3.3.66.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (1+\sqrt {x}-x\right )}{x} \, dx=\int \frac {\ln \left (\sqrt {x}-x+1\right )}{x} \,d x \]

input

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

output

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