\(\int \frac {\log (\frac {-b c+a d}{d (a+b x)}) \log (\frac {e (c+d x)}{a+b x})}{(a+b x) (c+d x)} \, dx\) [103]

Optimal result

Integrand size = 50, antiderivative size = 85 \[ \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d}-\frac {\operatorname {PolyLog}\left (3,1+\frac {b c-a d}{d (a+b x)}\right )}{b c-a d} \] Output:

ln(e*(d*x+c)/(b*x+a))*polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(-a*d+b*c)-polylog 
(3,1+(-a*d+b*c)/d/(b*x+a))/(-a*d+b*c)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.80 \[ \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )-\operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d} \] Input:

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

Output:

(Log[(e*(c + d*x))/(a + b*x)]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - Po 
lyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b*c - a*d)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2988, 7164}

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.

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

\(\Big \downarrow \) 2988

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

\(\Big \downarrow \) 7164

\(\displaystyle \frac {\operatorname {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{b c-a d}-\frac {\operatorname {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b c-a d}\)

Input:

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

Output:

(Log[(e*(c + d*x))/(a + b*x)]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/( 
b*c - a*d) - PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))]/(b*c - a*d)
 

Defintions of rubi rules used

rule 2988

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_) 
)^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> With[{g = Simplify[(v - 1)*((c + d 
*x)/(a + b*x))], h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(-h)*PolyLog[2, 
 1 - v]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d)), x] + Simp[h*p 
*r*s   Int[PolyLog[2, 1 - v]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/ 
((a + b*x)*(c + d*x))), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b, c, d, e 
, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [A] (verified)

Time = 10.73 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.84

method	result	size
default	\(\frac {\frac {\ln \left (-\frac {\frac {e \left (d x +c \right ) b}{b x +a}-d e}{d e}\right ) \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2}}{2}-\frac {\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{2}-\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \operatorname {polylog}\left (2, \frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )+\operatorname {polylog}\left (3, \frac {b \left (d x +c \right )}{d \left (b x +a \right )}\right )}{d a -b c}\)	\(156\)

Input:

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

Output:

1/(a*d-b*c)*(1/2*ln(-(e*(d*x+c)/(b*x+a)*b-d*e)/d/e)*ln(e*(d*x+c)/(b*x+a))^ 
2-1/2*ln(e*(d*x+c)/(b*x+a))^2*ln(1-b*(d*x+c)/d/(b*x+a))-ln(e*(d*x+c)/(b*x+ 
a))*polylog(2,b*(d*x+c)/d/(b*x+a))+polylog(3,b*(d*x+c)/d/(b*x+a)))
 

Fricas [F]

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

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

Output:

integral(log(-(b*c - a*d)/(b*d*x + a*d))*log((d*e*x + c*e)/(b*x + a))/(b*d 
*x^2 + a*c + (b*c + a*d)*x), x)
 

Sympy [F]

\[ \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\frac {b \int \frac {\log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}^{2}}{a + b x}\, dx}{2 \left (a d - b c\right )} + \frac {\log {\left (\frac {a d - b c}{d \left (a + b x\right )} \right )} \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}^{2}}{2 a d - 2 b c} \] Input:

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

Output:

b*Integral(log(c*e/(a + b*x) + d*e*x/(a + b*x))**2/(a + b*x), x)/(2*(a*d - 
 b*c)) + log((a*d - b*c)/(d*(a + b*x)))*log(e*(c + d*x)/(a + b*x))**2/(2*a 
*d - 2*b*c)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\mathrm {log}\left (\frac {a d -b c}{b d x +a d}\right ) \mathrm {log}\left (\frac {d e x +c e}{b x +a}\right )}{b d \,x^{2}+a d x +b c x +a c}d x \] Input:

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

Output:

int((log((a*d - b*c)/(a*d + b*d*x))*log((c*e + d*e*x)/(a + b*x)))/(a*c + a 
*d*x + b*c*x + b*d*x**2),x)