Integrand size = 65, antiderivative size = 433 \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b c-a d)}-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{2 (b c-a d)}+\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (1-\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b c-a d)}-\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d}+\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d}+\frac {\operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d}-\frac {\operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d} \] Output:
-1/2*ln(-(-a*d+b*c)/d/(b*x+a))*ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2 /(-a*d+b*c)-ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2*ln(b*(f*x+e)/(-a*f +b*e))/(-2*a*d+2*b*c)+ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^2*ln(1-(-a *f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))/(-2*a*d+2*b*c)-ln((-a*f+b*e)*(d*x+c)/( -c*f+d*e)/(b*x+a))*polylog(2,b*(d*x+c)/d/(b*x+a))/(-a*d+b*c)+ln((-a*f+b*e) *(d*x+c)/(-c*f+d*e)/(b*x+a))*polylog(2,(-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+ a))/(-a*d+b*c)+polylog(3,b*(d*x+c)/d/(b*x+a))/(-a*d+b*c)-polylog(3,(-a*f+b *e)*(d*x+c)/(-c*f+d*e)/(b*x+a))/(-a*d+b*c)
Leaf count is larger than twice the leaf count of optimal. \(1855\) vs. \(2(433)=866\).
Time = 0.87 (sec) , antiderivative size = 1855, normalized size of antiderivative = 4.28 \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[(Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[(b*(e + f*x))/(b*e - a*f)])/((a + b*x)*(c + d*x)),x]
Output:
(2*Log[c/d + x]*Log[e/f + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[a/b + x]*Log[e/f + x]*Log[(b*(c + d*x))/(b*c - a*d)] - 2*(Log[a + b*x] - Log[ c + d*x])*(Log[a/b + x] - Log[c/d + x] + Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])*(Log[e/f + x] - Log[(b*(e + f*x))/(b*e - a*f)]) + (Lo g[(d*(a + b*x))/(-(b*c) + a*d)] - Log[(f*(a + b*x))/(-(b*e) + a*f)])*Log[( b*(e + f*x))/(b*e - a*f)]*(-2*Log[c/d + x] + Log[(b*(e + f*x))/(b*e - a*f) ]) + Log[a/b + x]^2*(-Log[e/f + x] + Log[(b*(e + f*x))/(b*e - a*f)]) + (Lo g[(b*(c + d*x))/(b*c - a*d)] - Log[(f*(c + d*x))/(-(d*e) + c*f)])*Log[(d*( e + f*x))/(d*e - c*f)]*(-2*Log[a/b + x] + Log[(d*(e + f*x))/(d*e - c*f)]) + Log[c/d + x]^2*(-Log[e/f + x] + Log[(d*(e + f*x))/(d*e - c*f)]) + 2*(-Lo g[(b*(c + d*x))/(b*c - a*d)] + Log[(f*(c + d*x))/(-(d*e) + c*f)])*Log[(d*( e + f*x))/(d*e - c*f)]*Log[((-(b*c) + a*d)*(e + f*x))/((d*e - c*f)*(a + b* x))] + (Log[(-(b*e) + a*f)/(f*(a + b*x))] + Log[(b*(c + d*x))/(b*c - a*d)] - Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])*Log[((-(b*c) + a* d)*(e + f*x))/((d*e - c*f)*(a + b*x))]^2 + 2*(-Log[(d*(a + b*x))/(-(b*c) + a*d)] + Log[(f*(a + b*x))/(-(b*e) + a*f)])*Log[(b*(e + f*x))/(b*e - a*f)] *Log[((b*c - a*d)*(e + f*x))/((b*e - a*f)*(c + d*x))] + (Log[(d*(a + b*x)) /(-(b*c) + a*d)] + Log[(-(d*e) + c*f)/(f*(c + d*x))] - Log[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])*Log[((b*c - a*d)*(e + f*x))/((b*e - a*f )*(c + d*x))]^2 + 2*(Log[e/f + x] - Log[((-(b*c) + a*d)*(e + f*x))/((d*...
Time = 1.47 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2989, 2954, 2804, 2009}
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.
| \(\displaystyle \int \frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{(a+b x) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2989 |
| \(\displaystyle \frac {f \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{e+f x}dx}{2 (b c-a d)}-\frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}\) |
| \(\Big \downarrow \) 2954 |
| \(\displaystyle -\frac {1}{2} f \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{\left (d-\frac {b (c+d x)}{a+b x}\right ) \left (d e-c f-\frac {(b e-a f) (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}-\frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle -\frac {1}{2} f \int \left (\frac {b \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(b c-a d) f \left (\frac {b (c+d x)}{a+b x}-d\right )}+\frac {(b e-a f) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(b c-a d) f \left (d e-c f-\frac {(b e-a f) (c+d x)}{a+b x}\right )}\right )d\frac {c+d x}{a+b x}-\frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} f \left (\frac {2 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f (b c-a d)}+\frac {2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f (b c-a d)}-\frac {2 \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f (b c-a d)}+\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f (b c-a d)}-\frac {\log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f (b c-a d)}-\frac {2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{f (b c-a d)}\right )-\frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}\) |
Input:
Int[(Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[(b*(e + f*x) )/(b*e - a*f)])/((a + b*x)*(c + d*x)),x]
Output:
-1/2*(Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[(b*(e + f *x))/(b*e - a*f)])/(b*c - a*d) - (f*((Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/((b*c - a*d)*f) - (Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[1 - ((b*e - a *f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((b*c - a*d)*f) + (2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/((b*c - a*d)*f) - (2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((b*c - a*d)*f) - (2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/((b*c - a*d)*f) + (2*PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((b*c - a *d)*f)))/2
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d) Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B , n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m ] && IGtQ[p, 0]
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_S ymbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[k*Log[i*(j*(g + h*x)^t)^u]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(p*r*(s + 1)*(b*c - a*d))), x] - Simp[k*h*t*(u/(p*r*(s + 1)*(b*c - a*d))) Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]] /; FreeQ[ {a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] & & EqQ[p + q, 0] && NeQ[s, -1]
\[\int \frac {\ln \left (\frac {\left (-f a +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right ) \ln \left (\frac {b \left (f x +e \right )}{-f a +b e}\right )}{\left (b x +a \right ) \left (d x +c \right )}d x\]
Input:
int(ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*ln(b*(f*x+e)/(-a*f+b*e))/(b* x+a)/(d*x+c),x)
Output:
int(ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*ln(b*(f*x+e)/(-a*f+b*e))/(b* x+a)/(d*x+c),x)
\[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left (\frac {{\left (f x + e\right )} b}{b e - a f}\right ) \log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \] Input:
integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*log(b*(f*x+e)/(-a*f+b *e))/(b*x+a)/(d*x+c),x, algorithm="fricas")
Output:
integral(log((b*c*e - a*c*f + (b*d*e - a*d*f)*x)/(a*d*e - a*c*f + (b*d*e - b*c*f)*x))*log((b*f*x + b*e)/(b*e - a*f))/(b*d*x^2 + a*c + (b*c + a*d)*x) , x)
Timed out. \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(ln((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*ln(b*(f*x+e)/(-a*f+b*e ))/(b*x+a)/(d*x+c),x)
Output:
Timed out
Exception generated. \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*log(b*(f*x+e)/(-a*f+b *e))/(b*x+a)/(d*x+c),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: Memory limit reached. Please j ump to an outer pointer, quit program and enlarge thememory limits before executing the program again.
\[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left (\frac {{\left (f x + e\right )} b}{b e - a f}\right ) \log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \] Input:
integrate(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*log(b*(f*x+e)/(-a*f+b *e))/(b*x+a)/(d*x+c),x, algorithm="giac")
Output:
integrate(log((f*x + e)*b/(b*e - a*f))*log((b*e - a*f)*(d*x + c)/((d*e - c *f)*(b*x + a)))/((b*x + a)*(d*x + c)), x)
Timed out. \[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\ln \left (-\frac {b\,\left (e+f\,x\right )}{a\,f-b\,e}\right )\,\ln \left (\frac {\left (a\,f-b\,e\right )\,\left (c+d\,x\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \] Input:
int((log(-(b*(e + f*x))/(a*f - b*e))*log(((a*f - b*e)*(c + d*x))/((c*f - d *e)*(a + b*x))))/((a + b*x)*(c + d*x)),x)
Output:
int((log(-(b*(e + f*x))/(a*f - b*e))*log(((a*f - b*e)*(c + d*x))/((c*f - d *e)*(a + b*x))))/((a + b*x)*(c + d*x)), x)
\[ \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\mathrm {log}\left (\frac {-b f x -b e}{a f -b e}\right ) \mathrm {log}\left (\frac {a d f x -b d e x +a c f -b c e}{b c f x -b d e x +a c f -a d e}\right )}{b d \,x^{2}+a d x +b c x +a c}d x \] Input:
int(log((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))*log(b*(f*x+e)/(-a*f+b*e))/( b*x+a)/(d*x+c),x)
Output:
int((log(( - b*e - b*f*x)/(a*f - b*e))*log((a*c*f + a*d*f*x - b*c*e - b*d* e*x)/(a*c*f - a*d*e + b*c*f*x - b*d*e*x)))/(a*c + a*d*x + b*c*x + b*d*x**2 ),x)