3.2.6 \(\int \frac {\log (\frac {d (a+b x)}{b (c+d x)})}{c f+d f x} \, dx\) [106]

3.2.6.1 Optimal result

Integrand size = 29, antiderivative size = 28 \[ \int \frac {\log \left (\frac {d (a+b x)}{b (c+d x)}\right )}{c f+d f x} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {b c-a d}{b (c+d x)}\right )}{d f} \]

output

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

3.2.6.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(28)=56\).

Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.07 \[ \int \frac {\log \left (\frac {d (a+b x)}{b (c+d x)}\right )}{c f+d f x} \, dx=\frac {\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-2 \log \left (\frac {d (a+b x)}{b (c+d x)}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 d f} \]

input

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

output

(Log[(b*c - a*d)/(b*c + b*d*x)]*(2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - 2*L 
og[(d*(a + b*x))/(b*(c + d*x))] + Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*Poly 
Log[2, (b*(c + d*x))/(b*c - a*d)])/(2*d*f)
 

3.2.6.3 Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2897}

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.

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

\(\Big \downarrow \) 2897

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

input

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

output

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

3.2.6.3.1 Defintions of rubi rules used

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

3.2.6.4 Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {\operatorname {dilog}\left (1+\frac {a d -c b}{b \left (d x +c \right )}\right )}{d f}\)	\(30\)
default	\(\frac {\operatorname {dilog}\left (1+\frac {a d -c b}{b \left (d x +c \right )}\right )}{d f}\)	\(30\)
risch	\(\frac {\operatorname {dilog}\left (1+\frac {a d -c b}{b \left (d x +c \right )}\right )}{d f}\)	\(30\)
parts	\(\frac {\ln \left (\frac {d \left (b x +a \right )}{b \left (d x +c \right )}\right ) \ln \left (d x +c \right )}{d f}-\frac {b \left (-\frac {d^{2} \ln \left (d x +c \right )^{2}}{2 b}+d^{2} \left (\frac {\operatorname {dilog}\left (\frac {a d -c b +b \left (d x +c \right )}{a d -c b}\right )}{b}+\frac {\ln \left (d x +c \right ) \ln \left (\frac {a d -c b +b \left (d x +c \right )}{a d -c b}\right )}{b}\right )\right )}{d^{3} f}\)	\(132\)

input

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

output

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

3.2.6.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (\frac {d (a+b x)}{b (c+d x)}\right )}{c f+d f x} \, dx=\frac {{\rm Li}_2\left (-\frac {b d x + a d}{b d x + b c} + 1\right )}{d f} \]

input

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

output

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

3.2.6.6 Sympy [F]

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

input

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

output

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

3.2.6.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (27) = 54\).

Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 5.64 \[ \int \frac {\log \left (\frac {d (a+b x)}{b (c+d x)}\right )}{c f+d f x} \, dx=-\frac {b {\left (\frac {\log \left (d x + c\right )^{2}}{b f} - \frac {2 \, {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b f}\right )}}{2 \, d} - \frac {b {\left (\frac {d \log \left (b x + a\right )}{b} - \frac {d \log \left (d x + c\right )}{b}\right )} \log \left (d f x + c f\right )}{d^{2} f} + \frac {\log \left (d f x + c f\right ) \log \left (\frac {{\left (b x + a\right )} d}{{\left (d x + c\right )} b}\right )}{d f} \]

input

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

output

-1/2*b*(log(d*x + c)^2/(b*f) - 2*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a* 
d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))/(b*f))/d - b*(d*log(b*x + a)/ 
b - d*log(d*x + c)/b)*log(d*f*x + c*f)/(d^2*f) + log(d*f*x + c*f)*log((b*x 
 + a)*d/((d*x + c)*b))/(d*f)
 

3.2.6.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1203 vs. \(2 (27) = 54\).

Time = 34.96 (sec) , antiderivative size = 1203, normalized size of antiderivative = 42.96 \[ \int \frac {\log \left (\frac {d (a+b x)}{b (c+d x)}\right )}{c f+d f x} \, dx=\text {Too large to display} \]

input

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

output

-1/2*(b^2*c*d/(b*c - a*d)^2 - a*b*d^2/(b*c - a*d)^2)^2*((b^3*c^3 - 3*a*b^2 
*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(log(abs(b*d*x + a*d)/abs(b*d*x + b*c))/ 
(b^3*d^4*f) - log(abs((b*d*x + a*d)/(b*d*x + b*c) - 1))/(b^3*d^4*f) - 1/(b 
^3*d^4*f*((b*d*x + a*d)/(b*d*x + b*c) - 1))) - (b^3*c^3 - 3*a*b^2*c^2*d + 
3*a^2*b*c*d^2 - a^3*d^3)*log((a + b*((a*d - b*((b*d*x + a*d)*b*c/((b*d*x + 
 b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)* 
(b*c - a*d)) - b*d/(b*c - a*d)))*b*c/((b*c - a*d)*(b*c - b*((b*d*x + a*d)* 
b*c/((b*d*x + b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/(( 
b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))) - a*d/(b*c - a*d))/(b*d/(b* 
c - a*d) - (a*d - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d)) - a*d/( 
b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - 
a*d)))*b*d/((b*c - a*d)*(b*c - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - 
a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) 
- b*d/(b*c - a*d))))))*d/(b*(c + ((a*d - b*((b*d*x + a*d)*b*c/((b*d*x + b* 
c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b* 
c - a*d)) - b*d/(b*c - a*d)))*b*c/((b*c - a*d)*(b*c - b*((b*d*x + a*d)*b*c 
/((b*d*x + b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d 
*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))) - a*d/(b*c - a*d))*d/(b*d/(b*c 
 - a*d) - (a*d - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d)) - a*d/(b 
*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c ...
 

3.2.6.9 Mupad [B] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\log \left (\frac {d (a+b x)}{b (c+d x)}\right )}{c f+d f x} \, dx=\frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {d\,\left (a+b\,x\right )}{b\,\left (c+d\,x\right )}\right )}{d\,f} \]

input

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

output

dilog((d*(a + b*x))/(b*(c + d*x)))/(d*f)