Integrand size = 43, antiderivative size = 371 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{2 (b c-a d) n}+\frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{2 (b c-a d) n}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b c-a d) n}+\frac {m \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {m \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {m n \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {m n \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d} \]
1/2*m*ln(e*((b*x+a)/(d*x+c))^n)^2*ln((-a*d+b*c)/b/(d*x+c))/(-a*d+b*c)/n+1/ 2*ln(e*((b*x+a)/(d*x+c))^n)^2*ln(h*(g*x+f)^m)/(-a*d+b*c)/n-1/2*m*ln(e*((b* x+a)/(d*x+c))^n)^2*ln(1-(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)/ n+m*ln(e*((b*x+a)/(d*x+c))^n)*polylog(2,d*(b*x+a)/b/(d*x+c))/(-a*d+b*c)-m* ln(e*((b*x+a)/(d*x+c))^n)*polylog(2,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c)) /(-a*d+b*c)-m*n*polylog(3,d*(b*x+a)/b/(d*x+c))/(-a*d+b*c)+m*n*polylog(3,(- c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)
Leaf count is larger than twice the leaf count of optimal. \(1842\) vs. \(2(371)=742\).
Time = 0.88 (sec) , antiderivative size = 1842, normalized size of antiderivative = 4.96 \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx =\text {Too large to display} \]
(m*n*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]^2 - 2*m*n*Log[a/b + x]^2*Log[f + g*x] + 2*m*n*Log[a/b + x]*Log[c/d + x]*Log[f + g*x] - 2*m*n*Log[c/d + x]^2*Log[f + g*x] + 2*m*n*L og[a/b + x]*Log[a + b*x]*Log[f + g*x] - 2*m*n*Log[c/d + x]*Log[a + b*x]*Lo g[f + g*x] + 2*m*n*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[f + g*x] + 2*m*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] - 2*m* Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] - 2*m*Log[a + b*x ]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] - 2*m*n*Log[a/b + x]*Log[c + d*x]*Log[f + g*x] + 2*m*n*Log[c/d + x]*Log[c + d*x]*Log[f + g*x] + 2*m*Lo g[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x]*Log[f + g*x] + 2*m*n*Log[a/b + x ]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[f + g*x] + m*n*Log[a/b + x]^2*Log[(b* (f + g*x))/(b*f - a*g)] - 2*m*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]* Log[(b*(f + g*x))/(b*f - a*g)] - 2*m*n*Log[a/b + x]*Log[(g*(c + d*x))/(-(d *f) + c*g)]*Log[(b*(f + g*x))/(b*f - a*g)] + m*n*Log[(g*(c + d*x))/(-(d*f) + c*g)]^2*Log[(b*(f + g*x))/(b*f - a*g)] - 2*m*n*Log[(g*(c + d*x))/(-(d*f ) + c*g)]*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*Log[(b*(f + g*x))/(b*f - a*g)] + m*n*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b* x))]^2*Log[(b*(f + g*x))/(b*f - a*g)] - 2*m*n*Log[a/b + x]*Log[c/d + x]*Lo g[(d*(f + g*x))/(d*f - c*g)] + m*n*Log[c/d + x]^2*Log[(d*(f + g*x))/(d*f - c*g)] + 2*m*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(d*(f + g*...
Time = 0.72 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {2989, 2953, 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.
| \(\displaystyle \int \frac {\log \left (h (f+g x)^m\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2989 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}-\frac {g m \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x}dx}{2 n (b c-a d)}\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}-\frac {g m \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right ) \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{2 n}\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}-\frac {g m \int \left (\frac {d \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) g \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(c g-d f) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) g \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}}{2 n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}-\frac {g m \left (\frac {2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g (b c-a d)}+\frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{g (b c-a d)}-\frac {2 n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g (b c-a d)}+\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g (b c-a d)}\right )}{2 n}\) |
(Log[e*((a + b*x)/(c + d*x))^n]^2*Log[h*(f + g*x)^m])/(2*(b*c - a*d)*n) - (g*m*(-((Log[e*((a + b*x)/(c + d*x))^n]^2*Log[1 - (d*(a + b*x))/(b*(c + d* x))])/((b*c - a*d)*g)) + (Log[e*((a + b*x)/(c + d*x))^n]^2*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g) - (2*n*Log[e*(( a + b*x)/(c + d*x))^n]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a* d)*g) + (2*n*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[2, ((d*f - c*g)*(a + b *x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g) + (2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a*d)*g) - (2*n^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g)))/(2*n)
3.1.69.3.1 Defintions of rubi rules used
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_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d) Sub st[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] && 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]
\[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right )}d x\]
\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
integral(log((g*x + f)^m*h)*log(e*((b*x + a)/(d*x + c))^n)/(b*d*x^2 + a*c + (b*c + a*d)*x), x)
Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
-1/2*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log(b*x + a) - log(e))*lo g(d*x + c) - 2*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n) + 2*(log(b*x + a) - log(d*x + c))*log((d*x + c)^n) - 2*log(b*x + a)*log(e))*log((g*x + f)^m)/(b*c - a*d) + integrate(1/2*(2*b*c*f*log(e)*log(h) - 2*a*d*f*log(e) *log(h) + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a)^2 + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(d*x + c)^2 + 2*(b*c*g*log(e)*log(h) - a*d*g*log(e)*log(h))*x - 2*(b*d*g*m*x^2*l og(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x)*log(b*x + a) + 2*(b*d*g*m*x^2*log(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log( e))*x - (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a))*log(d*x + c) + 2*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g* log(h))*x - (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((b*x + a)^n) - 2*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x ^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((d*x + c)^n))/(a*b *c^2*f - a^2*c*d*f + (b^2*c*d*g - a*b*d^2*g)*x^3 - (a*b*d^2*f + a^2*d^2*g - (c*d*f + c^2*g)*b^2)*x^2 + (b^2*c^2*f + a*b*c^2*g - (d^2*f + c*d*g)*a^2) *x), x)
\[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]