3.1.64 \(\int \frac {\log (x) \log (\frac {a+b x}{(b c-a d) x})}{x} \, dx\) [64]

3.1.64.1 Optimal result

Integrand size = 26, antiderivative size = 82 \[ \int \frac {\log (x) \log \left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=-\frac {1}{2} \log \left (1+\frac {a}{b x}\right ) \log ^2(x)+\frac {1}{2} \log \left (\frac {b}{b c-a d}+\frac {a}{(b c-a d) x}\right ) \log ^2(x)+\log (x) \operatorname {PolyLog}\left (2,-\frac {a}{b x}\right )+\operatorname {PolyLog}\left (3,-\frac {a}{b x}\right ) \]

output

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

3.1.64.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \frac {\log (x) \log \left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\frac {1}{6} \log ^2(x) \left (\log (x)-3 \log \left (1+\frac {b x}{a}\right )+3 \log \left (\frac {a+b x}{b c x-a d x}\right )\right )-\log (x) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )+\operatorname {PolyLog}\left (3,-\frac {b x}{a}\right ) \]

input

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

output

(Log[x]^2*(Log[x] - 3*Log[1 + (b*x)/a] + 3*Log[(a + b*x)/(b*c*x - a*d*x)]) 
)/6 - Log[x]*PolyLog[2, -((b*x)/a)] + PolyLog[3, -((b*x)/a)]
 

3.1.64.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2827, 2822, 27, 2005, 2779, 2821, 7143}

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.

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

\(\Big \downarrow \) 2827

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

\(\Big \downarrow \) 2822

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.1.64.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2822

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_ 
.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp[Log[d*(e + f*x^m)^r]*((a + b*Log[ 
c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[f*m*(r/(b*n*(p + 1)))   Int[x^(m 
- 1)*((a + b*Log[c*x^n])^(p + 1)/(e + f*x^m)), x], x] /; FreeQ[{a, b, c, d, 
 e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[d*e, 1]
 

rule 2827

Int[Log[(d_.)*(u_)^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_. 
)*(x_))^(q_.), x_Symbol] :> Int[(g*x)^q*Log[d*ExpandToSum[u, x]^r]*(a + b*L 
og[c*x^n])^p, x] /; FreeQ[{a, b, c, d, g, r, n, p, q}, x] && BinomialQ[u, x 
] &&  !BinomialMatchQ[u, x]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.1.64.4 Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {\ln \left (-\frac {b x +a}{\left (a d -c b \right ) x}\right ) \ln \left (x \right )^{2}}{2}+\frac {\left (-\frac {a d}{2}+\frac {c b}{2}\right ) \left (-\frac {\ln \left (x \right )^{3}}{3}+\ln \left (x \right )^{2} \ln \left (1+\frac {x b}{a}\right )+2 \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {x b}{a}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {x b}{a}\right )\right )}{a d -c b}\)	\(94\)
risch	\(\frac {\ln \left (x \right )^{2} \ln \left (b x +a \right )}{2}-\frac {\ln \left (x \right )^{3}}{3}-\frac {\left (2 i \pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{x \left (a d -c b \right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )\right ) \operatorname {csgn}\left (\frac {i}{a d -c b}\right ) \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{a d -c b}\right )-i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )\right ) \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{a d -c b}\right )^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{a d -c b}\right ) \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{a d -c b}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{a d -c b}\right ) \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{x \left (a d -c b \right )}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{x \left (a d -c b \right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{a d -c b}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (b x +a \right )}{a d -c b}\right ) \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{x \left (a d -c b \right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (b x +a \right )}{x \left (a d -c b \right )}\right )^{3}-2 i \pi +2 \ln \left (a d -c b \right )\right ) \ln \left (x \right )^{2}}{4}-\frac {\ln \left (x \right )^{2} \ln \left (1+\frac {x b}{a}\right )}{2}-\ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {x b}{a}\right )+\operatorname {Li}_{3}\left (-\frac {x b}{a}\right )\)	\(413\)

input

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

output

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

3.1.64.5 Fricas [F]

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

input

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

output

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

3.1.64.6 Sympy [F]

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

input

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

output

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

3.1.64.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\log (x) \log \left (\frac {a+b x}{(b c-a d) x}\right )}{x} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> unable to make sense of Maxima expression ' 
li[2]' in Sage
 

3.1.64.8 Giac [F]

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

input

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

output

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

3.1.64.9 Mupad [F(-1)]

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

input

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

output

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