\(\int \frac {\log (\frac {e (c+d x)}{a+b x}) \log (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)})}{(a+b x) (c+d x)} \, dx\) [105]

Optimal result

Integrand size = 62, antiderivative size = 109 \[ \int \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \operatorname {PolyLog}\left (2,1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d}-\frac {\operatorname {PolyLog}\left (3,1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2588, 6745} \[ \int \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \operatorname {PolyLog}\left (2,\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1\right )}{b c-a d}-\frac {\operatorname {PolyLog}\left (3,\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1\right )}{b c-a d} \]

[In]

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

[Out]

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

Rule 2588

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[(v - 1)*((c + d*x)/(a + b*x))], h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(-h)*PolyL
og[2, 1 - v]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d)), x] + Dist[h*p*r*s, Int[PolyLog[2, 1 - v]*(L
og[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 6745

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d}+\int \frac {\text {Li}_2\left (1-\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx \\ & = \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d}-\frac {\text {Li}_3\left (1+\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{b c-a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \log \left (\frac {(-b c+a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx=\frac {\log \left (\frac {e (c+d x)}{a+b x}\right ) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-\operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(422\) vs. \(2(109)=218\).

Time = 25.00 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.88

method	result	size
default	\(\frac {\frac {\ln \left (-\frac {\frac {e \left (d x +c \right ) a f}{b x +a}-\frac {e^{2} \left (d x +c \right ) b}{b x +a}-c e f +d \,e^{2}}{e \left (c f -d e \right )}\right ) \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2}}{2}-\frac {a f \left (\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )+2 \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \operatorname {Li}_{2}\left (\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )-2 \,\operatorname {Li}_{3}\left (\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )\right )}{2 \left (a f -b e \right )}+\frac {b e \left (\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} \ln \left (1-\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )+2 \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) \operatorname {Li}_{2}\left (\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )-2 \,\operatorname {Li}_{3}\left (\frac {\left (a f -b e \right ) e \left (d x +c \right )}{\left (b x +a \right ) \left (c e f -d \,e^{2}\right )}\right )\right )}{2 a f -2 b e}}{a d -c b}\)	\(423\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/2*(log(b*x + a)^2 - 2*(log(b*x + a) - log(e))*log(d*x + c) + log(d*x + c)^2 - 2*log(b*x + a)*log(e))*log(f*
x + e)/(b*c - a*d) + integrate(1/2*(2*(e*log(-b*c + a*d)*log(e) - e*log(d*e - c*f)*log(e))*b*c + (b*d*f*x^2 +
2*b*c*e - (2*d*e - c*f)*a + (3*b*c*f - a*d*f)*x)*log(b*x + a)^2 - 2*(d*e*log(-b*c + a*d)*log(e) - d*e*log(d*e
- c*f)*log(e))*a + 2*((f*log(-b*c + a*d)*log(e) - f*log(d*e - c*f)*log(e))*b*c - (d*f*log(-b*c + a*d)*log(e) -
d*f*log(d*e - c*f)*log(e))*a)*x - 2*(b*d*f*x^2*log(e) - (e*(log(d*e - c*f) - log(e)) - e*log(-b*c + a*d))*b*c
+ (d*e*(log(d*e - c*f) - log(e)) - d*e*log(-b*c + a*d) + c*f*log(e))*a + ((f*log(-b*c + a*d) - f*log(d*e - c*
f) + 2*f*log(e))*b*c - (d*f*log(-b*c + a*d) - d*f*log(d*e - c*f))*a)*x)*log(b*x + a) + 2*(b*d*f*x^2*log(e) + (
e*log(-b*c + a*d) - e*log(d*e - c*f))*b*c - (d*e*log(-b*c + a*d) - d*e*log(d*e - c*f) - c*f*log(e))*a + ((f*lo
g(-b*c + a*d) - f*log(d*e - c*f) + f*log(e))*b*c - (d*f*log(-b*c + a*d) - (f*log(d*e - c*f) + f*log(e))*d)*a)*
x - (b*d*f*x^2 + 2*b*c*f*x + b*c*e - (d*e - c*f)*a)*log(b*x + a))*log(d*x + c))/(a*b*c^2*e - a^2*c*d*e + (b^2*
c*d*f - a*b*d^2*f)*x^3 - (a*b*d^2*e + a^2*d^2*f - (c*d*e + c^2*f)*b^2)*x^2 + (b^2*c^2*e + a*b*c^2*f - (d^2*e +
c*d*f)*a^2)*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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