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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2512, 266, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {\log \left (a+\frac {b}{x}\right )}{c+d x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {a (c+d x)}{a c-b d}\right )}{d}+\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)}{d}-\frac {\log (c+d x) \log \left (-\frac {d (a x+b)}{a c-b d}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d} \]

[In]

Int[Log[a + b/x]/(c + d*x),x]

[Out]

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

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 2512

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)}{d}+\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}+\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}+\frac {\int \frac {\log (c+d x)}{x} \, dx}{d}-\frac {a \int \frac {\log (c+d x)}{b+a x} \, dx}{d} \\ & = \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}-\int \frac {\log \left (-\frac {d x}{c}\right )}{c+d x} \, dx+\int \frac {\log \left (\frac {d (b+a x)}{-a c+b d}\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 {\text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a c+b d}\right )}{x} \, dx,x,c+d x\right )}{d} \\ & = \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 {\text {Li}_2\left (\frac {a (c+d x)}{a c-b d}\right )}{d}+\frac {\text {Li}_2\left (1+\frac {d x}{c}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (a+\frac {b}{x}\right )}{c+d x} \, dx=\frac {\log \left (a+\frac {b}{x}\right ) \log (c+d x)+\log (x) \log (c+d x)-\log \left (\frac {b}{a}+x\right ) \log (c+d x)+\log \left (\frac {b}{a}+x\right ) \log \left (\frac {a (c+d x)}{a c-b d}\right )-\log (x) \log \left (1+\frac {d x}{c}\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+\operatorname {PolyLog}\left (2,\frac {d (b+a x)}{-a c+b d}\right )}{d} \]

[In]

Integrate[Log[a + b/x]/(c + d*x),x]

[Out]

(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

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(log(a + b/x)/(c + d*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (a+\frac {b}{x}\right )}{c+d x} \, dx=-\frac {\log \left (\frac {d x}{c} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{c}\right )}{d} + \frac {\log \left (a x + b\right ) \log \left (\frac {a d x + b d}{a c - b d} + 1\right ) + {\rm Li}_2\left (-\frac {a d x + b d}{a c - b d}\right )}{d} \]

[In]

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

[Out]

-(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]

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

[In]

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

[Out]

integrate(log(a + b/x)/(d*x + c), x)

Mupad [F(-1)]

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

[In]

int(log(a + b/x)/(c + d*x),x)

[Out]

int(log(a + b/x)/(c + d*x), x)