\(\int \frac {\log (x+\sqrt {1+x})}{x} \, dx\) [32]

Optimal result

Integrand size = 14, antiderivative size = 313 \[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{x} \, dx=\log \left (-1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{3-\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (1-\sqrt {1+x}\right )}{3-\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (1-\sqrt {1+x}\right )}{3+\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (1+\sqrt {1+x}\right )}{1-\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (1+\sqrt {1+x}\right )}{1+\sqrt {5}}\right ) \] Output:

ln(-1+(1+x)^(1/2))*ln(x+(1+x)^(1/2))+ln(1+(1+x)^(1/2))*ln(x+(1+x)^(1/2))-l 
n(-1+(1+x)^(1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(3-5^(1/2)))-ln(1+(1+x)^(1/ 
2))*ln((-1+5^(1/2)-2*(1+x)^(1/2))/(5^(1/2)+1))-ln(1+(1+x)^(1/2))*ln((-1-5^ 
(1/2)-2*(1+x)^(1/2))/(-5^(1/2)+1))-ln(-1+(1+x)^(1/2))*ln((1+5^(1/2)+2*(1+x 
)^(1/2))/(3+5^(1/2)))-polylog(2,2*(1-(1+x)^(1/2))/(3-5^(1/2)))-polylog(2,2 
*(1-(1+x)^(1/2))/(3+5^(1/2)))-polylog(2,2*(1+(1+x)^(1/2))/(-5^(1/2)+1))-po 
lylog(2,2*(1+(1+x)^(1/2))/(5^(1/2)+1))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.97 \[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{x} \, dx=\log \left (1-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\log \left (\frac {1}{2} \left (3-\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\log \left (\frac {1}{2} \left (3+\sqrt {5}\right )\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (1+\sqrt {1+x}\right )}{1-\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,\frac {1-\sqrt {5}+2 \sqrt {1+x}}{3-\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right ) \] Input:

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

Output:

Log[1 - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] + Log[1 + Sqrt[1 + x]]*Log[x + S 
qrt[1 + x]] - Log[(3 - Sqrt[5])/2]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - Log[ 
(1 + Sqrt[5])/2]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - Log[(3 + Sqrt[5])/2]*L 
og[1 + Sqrt[5] + 2*Sqrt[1 + x]] - Log[1 + Sqrt[1 + x]]*Log[-((1 + Sqrt[5] 
+ 2*Sqrt[1 + x])/(1 - Sqrt[5]))] - PolyLog[2, (2*(1 + Sqrt[1 + x]))/(1 - S 
qrt[5])] + PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(3 - Sqrt[5])] + PolyL 
og[2, -((1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + Sqrt[5]))] + PolyLog[2, (1 + Sq 
rt[5] + 2*Sqrt[1 + x])/(3 + Sqrt[5])]
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3010, 25, 3008, 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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3008

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2 \left (1-\sqrt {x+1}\right )}{3-\sqrt {5}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2 \left (1-\sqrt {x+1}\right )}{3+\sqrt {5}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {x+1}+1\right )}{1-\sqrt {5}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {x+1}+1\right )}{1+\sqrt {5}}\right )+\frac {1}{2} \log \left (\sqrt {x+1}-1\right ) \log \left (x+\sqrt {x+1}\right )+\frac {1}{2} \log \left (\sqrt {x+1}+1\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{2} \log \left (\sqrt {x+1}-1\right ) \log \left (\frac {2 \sqrt {x+1}-\sqrt {5}+1}{3-\sqrt {5}}\right )-\frac {1}{2} \log \left (\sqrt {x+1}+1\right ) \log \left (-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+\sqrt {5}}\right )-\frac {1}{2} \log \left (\sqrt {x+1}+1\right ) \log \left (-\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-\sqrt {5}}\right )-\frac {1}{2} \log \left (\sqrt {x+1}-1\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{3+\sqrt {5}}\right )\right )\)

Input:

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

Output:

2*((Log[-1 + Sqrt[1 + x]]*Log[x + Sqrt[1 + x]])/2 + (Log[1 + Sqrt[1 + x]]* 
Log[x + Sqrt[1 + x]])/2 - (Log[-1 + Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt 
[1 + x])/(3 - Sqrt[5])])/2 - (Log[1 + Sqrt[1 + x]]*Log[-((1 - Sqrt[5] + 2* 
Sqrt[1 + x])/(1 + Sqrt[5]))])/2 - (Log[1 + Sqrt[1 + x]]*Log[-((1 + Sqrt[5] 
 + 2*Sqrt[1 + x])/(1 - Sqrt[5]))])/2 - (Log[-1 + Sqrt[1 + x]]*Log[(1 + Sqr 
t[5] + 2*Sqrt[1 + x])/(3 + Sqrt[5])])/2 - PolyLog[2, (2*(1 - Sqrt[1 + x])) 
/(3 - Sqrt[5])]/2 - PolyLog[2, (2*(1 - Sqrt[1 + x]))/(3 + Sqrt[5])]/2 - Po 
lyLog[2, (2*(1 + Sqrt[1 + x]))/(1 - Sqrt[5])]/2 - PolyLog[2, (2*(1 + Sqrt[ 
1 + x]))/(1 + Sqrt[5])]/2)
 

Defintions of rubi rules used

rule 25

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

rule 2009

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

rule 3008

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\ln \left (-1+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (-1+\sqrt {1+x}\right ) \ln \left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}-3}\right )-\ln \left (-1+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}-3}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )+\ln \left (1+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (1+\sqrt {1+x}\right ) \ln \left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}+1}\right )-\ln \left (1+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{\sqrt {5}-1}\right )-\operatorname {dilog}\left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}+1}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{\sqrt {5}-1}\right )\)	\(252\)
default	\(\ln \left (-1+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (-1+\sqrt {1+x}\right ) \ln \left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}-3}\right )-\ln \left (-1+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}-3}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )+\ln \left (1+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (1+\sqrt {1+x}\right ) \ln \left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}+1}\right )-\ln \left (1+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{\sqrt {5}-1}\right )-\operatorname {dilog}\left (\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{\sqrt {5}+1}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{\sqrt {5}-1}\right )\)	\(252\)
parts	\(\ln \left (x \right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right ) \ln \left (x \right )+\operatorname {dilog}\left (\frac {1+\sqrt {1+x}}{\frac {1}{2}+\frac {\sqrt {5}}{2}}\right )+\ln \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1+\sqrt {1+x}}{\frac {1}{2}+\frac {\sqrt {5}}{2}}\right )+\left (\ln \left (\sqrt {1+x}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-\ln \left (\frac {\sqrt {1+x}-\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}\right )\right ) \ln \left (\frac {1-\sqrt {1+x}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}\right )-\operatorname {dilog}\left (\frac {\sqrt {1+x}-\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {3}{2}-\frac {\sqrt {5}}{2}}\right )-\ln \left (\sqrt {1+x}+\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \ln \left (x \right )+\operatorname {dilog}\left (\frac {1+\sqrt {1+x}}{\frac {1}{2}-\frac {\sqrt {5}}{2}}\right )+\ln \left (\sqrt {1+x}+\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \ln \left (\frac {1+\sqrt {1+x}}{\frac {1}{2}-\frac {\sqrt {5}}{2}}\right )+\left (\ln \left (\sqrt {1+x}+\frac {1}{2}+\frac {\sqrt {5}}{2}\right )-\ln \left (\frac {\sqrt {1+x}+\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}\right )\right ) \ln \left (\frac {1-\sqrt {1+x}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}\right )-\operatorname {dilog}\left (\frac {\sqrt {1+x}+\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}\right )\)	\(317\)

Input:

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

Output:

ln(-1+(1+x)^(1/2))*ln(x+(1+x)^(1/2))-ln(-1+(1+x)^(1/2))*ln((-1+5^(1/2)-2*( 
1+x)^(1/2))/(5^(1/2)-3))-ln(-1+(1+x)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/( 
3+5^(1/2)))-dilog((-1+5^(1/2)-2*(1+x)^(1/2))/(5^(1/2)-3))-dilog((1+5^(1/2) 
+2*(1+x)^(1/2))/(3+5^(1/2)))+ln(1+(1+x)^(1/2))*ln(x+(1+x)^(1/2))-ln(1+(1+x 
)^(1/2))*ln((-1+5^(1/2)-2*(1+x)^(1/2))/(5^(1/2)+1))-ln(1+(1+x)^(1/2))*ln(( 
1+5^(1/2)+2*(1+x)^(1/2))/(5^(1/2)-1))-dilog((-1+5^(1/2)-2*(1+x)^(1/2))/(5^ 
(1/2)+1))-dilog((1+5^(1/2)+2*(1+x)^(1/2))/(5^(1/2)-1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(log(sqrt(x + 1) + x)/x,x)