\(\int \frac {\log (a+\frac {b}{x})}{c+d x} \, dx\) [205]

Optimal result

Integrand size = 16, antiderivative size = 105 \[ \int \frac {\log \left (a+\frac {b}{x}\right )}{c+d x} \, dx=\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d (b+a x)}{a c-b d}\right ) \log (c+d x)}{d}-\frac {\operatorname {PolyLog}\left (2,\frac {a (c+d x)}{a c-b d}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{d} \] Output:

ln(a+b/x)*ln(d*x+c)/d+ln(-d*x/c)*ln(d*x+c)/d-ln(-d*(a*x+b)/(a*c-b*d))*ln(d 
*x+c)/d-polylog(2,a*(d*x+c)/(a*c-b*d))/d+polylog(2,1+d*x/c)/d
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (a+\frac {b}{x}\right )}{c+d x} \, dx=\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)+\log (x) \log (c+d x)-\log \left (\frac {b}{a}+x\right ) \log (c+d x)+\log \left (\frac {b}{a}+x\right ) \log \left (\frac {a (c+d x)}{a c-b d}\right )-\log (x) \log \left (1+\frac {d x}{c}\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+\operatorname {PolyLog}\left (2,\frac {d (b+a x)}{-a c+b d}\right )}{d} \] Input:

Integrate[Log[a + b/x]/(c + d*x),x]
 

Output:

(Log[a + b/x]*Log[c + d*x] + Log[x]*Log[c + d*x] - Log[b/a + x]*Log[c + d* 
x] + Log[b/a + x]*Log[(a*(c + d*x))/(a*c - b*d)] - Log[x]*Log[1 + (d*x)/c] 
 - PolyLog[2, -((d*x)/c)] + PolyLog[2, (d*(b + a*x))/(-(a*c) + b*d)])/d
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2912, 2005, 2863, 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.

\(\displaystyle \int \frac {\log \left (a+\frac {b}{x}\right )}{c+d x} \, dx\)

\(\Big \downarrow \) 2912

\(\displaystyle \frac {b \int \frac {\log (c+d x)}{\left (a+\frac {b}{x}\right ) x^2}dx}{d}+\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)}{d}\)

\(\Big \downarrow \) 2005

\(\displaystyle \frac {b \int \frac {\log (c+d x)}{x (b+a x)}dx}{d}+\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)}{d}\)

\(\Big \downarrow \) 2863

\(\displaystyle \frac {b \int \left (\frac {\log (c+d x)}{b x}-\frac {a \log (c+d x)}{b (b+a x)}\right )dx}{d}+\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (-\frac {\operatorname {PolyLog}\left (2,\frac {a (c+d x)}{a c-b d}\right )}{b}-\frac {\log (c+d x) \log \left (-\frac {d (a x+b)}{a c-b d}\right )}{b}+\frac {\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{b}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{b}\right )}{d}+\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)}{d}\)

Input:

Int[Log[a + b/x]/(c + d*x),x]
 

Output:

(Log[a + b/x]*Log[c + d*x])/d + (b*((Log[-((d*x)/c)]*Log[c + d*x])/b - (Lo 
g[-((d*(b + a*x))/(a*c - b*d))]*Log[c + d*x])/b - PolyLog[2, (a*(c + d*x)) 
/(a*c - b*d)]/b + PolyLog[2, 1 + (d*x)/c]/b))/d
 

Defintions of rubi rules used

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

rule 2009

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

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

rule 2912

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_. 
)*(x_)), x_Symbol] :> Simp[Log[f + g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x 
] - Simp[b*e*n*(p/g)   Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x] /; 
FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]
 

Maple [A] (verified)

Time = 6.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09

method	result	size
risch	\(-\frac {\ln \left (a +\frac {b}{x}\right ) \ln \left (-\frac {b}{a x}\right )}{d}-\frac {\operatorname {dilog}\left (-\frac {b}{a x}\right )}{d}+\frac {\operatorname {dilog}\left (\frac {-a c +b d +c \left (a +\frac {b}{x}\right )}{-a c +b d}\right )}{d}+\frac {\ln \left (a +\frac {b}{x}\right ) \ln \left (\frac {-a c +b d +c \left (a +\frac {b}{x}\right )}{-a c +b d}\right )}{d}\)	\(114\)
derivativedivides	\(-b \left (\frac {\operatorname {dilog}\left (-\frac {b}{a x}\right )+\ln \left (a +\frac {b}{x}\right ) \ln \left (-\frac {b}{a x}\right )}{d b}-\frac {\left (\frac {\operatorname {dilog}\left (\frac {-a c +b d +c \left (a +\frac {b}{x}\right )}{-a c +b d}\right )}{c}+\frac {\ln \left (a +\frac {b}{x}\right ) \ln \left (\frac {-a c +b d +c \left (a +\frac {b}{x}\right )}{-a c +b d}\right )}{c}\right ) c}{d b}\right )\)	\(126\)
default	\(-b \left (\frac {\operatorname {dilog}\left (-\frac {b}{a x}\right )+\ln \left (a +\frac {b}{x}\right ) \ln \left (-\frac {b}{a x}\right )}{d b}-\frac {\left (\frac {\operatorname {dilog}\left (\frac {-a c +b d +c \left (a +\frac {b}{x}\right )}{-a c +b d}\right )}{c}+\frac {\ln \left (a +\frac {b}{x}\right ) \ln \left (\frac {-a c +b d +c \left (a +\frac {b}{x}\right )}{-a c +b d}\right )}{c}\right ) c}{d b}\right )\)	\(126\)
parts	\(\frac {\ln \left (a +\frac {b}{x}\right ) \ln \left (d x +c \right )}{d}+b \left (\frac {\operatorname {dilog}\left (-\frac {d x}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {d x}{c}\right )}{b d}-\frac {\left (\frac {\operatorname {dilog}\left (\frac {-a c +a \left (d x +c \right )+b d}{-a c +b d}\right )}{a}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {-a c +a \left (d x +c \right )+b d}{-a c +b d}\right )}{a}\right ) a}{b d}\right )\)	\(132\)

Input:

int(ln(a+b/x)/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/d*ln(a+b/x)*ln(-b/a/x)-1/d*dilog(-b/a/x)+1/d*dilog((-a*c+b*d+c*(a+b/x)) 
/(-a*c+b*d))+1/d*ln(a+b/x)*ln((-a*c+b*d+c*(a+b/x))/(-a*c+b*d))
 

Fricas [F]

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

integrate(log(a+b/x)/(d*x+c),x, algorithm="fricas")
 

Output:

integral(log((a*x + b)/x)/(d*x + c), x)
 

Sympy [F]

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

integrate(ln(a+b/x)/(d*x+c),x)
 

Output:

Integral(log(a + b/x)/(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (a+\frac {b}{x}\right )}{c+d x} \, dx=-\frac {\log \left (\frac {d x}{c} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{c}\right )}{d} + \frac {\log \left (a x + b\right ) \log \left (\frac {a d x + b d}{a c - b d} + 1\right ) + {\rm Li}_2\left (-\frac {a d x + b d}{a c - b d}\right )}{d} \] Input:

integrate(log(a+b/x)/(d*x+c),x, algorithm="maxima")
 

Output:

-(log(d*x/c + 1)*log(x) + dilog(-d*x/c))/d + (log(a*x + b)*log((a*d*x + b* 
d)/(a*c - b*d) + 1) + dilog(-(a*d*x + b*d)/(a*c - b*d)))/d
 

Giac [F]

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

integrate(log(a+b/x)/(d*x+c),x, algorithm="giac")
 

Output:

integrate(log(a + b/x)/(d*x + c), x)
 

Mupad [F(-1)]

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

int(log(a + b/x)/(c + d*x),x)
 

Output:

int(log(a + b/x)/(c + d*x), x)
 

Reduce [F]

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

int(log(a+b/x)/(d*x+c),x)
                                                                                    
                                                                                    
 

Output:

int(log((a*x + b)/x)/(c + d*x),x)