Integrand size = 26, antiderivative size = 71 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx=\frac {b n \log (x)}{e m}-\frac {b n \left (d+e \log \left (f x^m\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m} \]
b*n*ln(x)/e/m-b*n*(d+e*ln(f*x^m))*ln(d+e*ln(f*x^m))/e^2/m^2+(a+b*ln(c*x^n) )*ln(d+e*ln(f*x^m))/e/m
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx=\frac {b e m n \log (x)+\left (a e m-b d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2} \]
(b*e*m*n*Log[x] + (a*e*m - b*d*n - b*e*n*Log[f*x^m] + b*e*m*Log[c*x^n])*Lo g[d + e*Log[f*x^m]])/(e^2*m^2)
Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2813, 27, 3039, 2836, 2732}
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.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-b n \int \frac {\log \left (d+e \log \left (f x^m\right )\right )}{e m x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {b n \int \frac {\log \left (d+e \log \left (f x^m\right )\right )}{x}dx}{e m}\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {b n \int \log \left (d+e \log \left (f x^m\right )\right )d\log \left (f x^m\right )}{e m^2}\) |
\(\Big \downarrow \) 2836 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {b n \int \log \left (d+e \log \left (f x^m\right )\right )d\left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}\) |
\(\Big \downarrow \) 2732 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {b n \left (\left (d+e \log \left (f x^m\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )-d-e \log \left (f x^m\right )\right )}{e^2 m^2}\) |
((a + b*Log[c*x^n])*Log[d + e*Log[f*x^m]])/(e*m) - (b*n*(-d - e*Log[f*x^m] + (d + e*Log[f*x^m])*Log[d + e*Log[f*x^m]]))/(e^2*m^2)
3.2.73.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x ] /; FreeQ[{c, n}, x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r]) u, x] - Simp[e*r Int[Simp lifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.18 (sec) , antiderivative size = 1239, normalized size of antiderivative = 17.45
1/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c *x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c )+2*a)/m*ln(e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I *f*x^m)^2-e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*e*ln(f )+2*I*ln(x^m)*e+2*I*d)/e+b*n*ln(x)/e/m+1/2*I*b/e/m^2*ln(e*Pi*csgn(I*f)*csg n(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f*x^m)^2-e*Pi*csgn(I*x^m)*csg n(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*ln(x)*e*m+2*I*e*ln(f)+2*I*e*(ln(x^m) -m*ln(x))+2*I*d)*Pi*n*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/2*I*b/e/m^2*ln (e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f*x^m)^2-e *Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*ln(x)*e*m+2*I*e*l n(f)+2*I*e*(ln(x^m)-m*ln(x))+2*I*d)*Pi*n*csgn(I*f)*csgn(I*f*x^m)^2-1/2*I*b /e/m^2*ln(e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I*f *x^m)^2-e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*ln(x)*e* m+2*I*e*ln(f)+2*I*e*(ln(x^m)-m*ln(x))+2*I*d)*Pi*n*csgn(I*x^m)*csgn(I*f*x^m )^2+1/2*I*b/e/m^2*ln(e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I* f)*csgn(I*f*x^m)^2-e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2 *I*ln(x)*e*m+2*I*e*ln(f)+2*I*e*(ln(x^m)-m*ln(x))+2*I*d)*Pi*n*csgn(I*f*x^m) ^3+b/e/m*ln(e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-e*Pi*csgn(I*f)*csgn(I *f*x^m)^2-e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+e*Pi*csgn(I*f*x^m)^3+2*I*ln(x)* e*m+2*I*e*ln(f)+2*I*e*(ln(x^m)-m*ln(x))+2*I*d)*ln(x^n)-b/e/m^2*ln(e*Pi*...
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx=\frac {b e m n \log \left (x\right ) + {\left (b e m \log \left (c\right ) - b e n \log \left (f\right ) + a e m - b d n\right )} \log \left (e m \log \left (x\right ) + e \log \left (f\right ) + d\right )}{e^{2} m^{2}} \]
(b*e*m*n*log(x) + (b*e*m*log(c) - b*e*n*log(f) + a*e*m - b*d*n)*log(e*m*lo g(x) + e*log(f) + d))/(e^2*m^2)
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x \left (d + e \log {\left (f x^{m} \right )}\right )}\, dx \]
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.66 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx=\frac {b \log \left (c x^{n}\right ) \log \left (\frac {e \log \left (f\right ) + e \log \left (x^{m}\right ) + d}{e}\right )}{e m} - \frac {b n {\left (\frac {{\left (e \log \left (f\right ) + e \log \left (x^{m}\right ) + d\right )} \log \left (\frac {e \log \left (f\right ) + e \log \left (x^{m}\right ) + d}{e}\right )}{e} - \frac {e \log \left (f\right ) + e \log \left (x^{m}\right ) + d}{e}\right )}}{e m^{2}} + \frac {a \log \left (\frac {e \log \left (f\right ) + e \log \left (x^{m}\right ) + d}{e}\right )}{e m} \]
b*log(c*x^n)*log((e*log(f) + e*log(x^m) + d)/e)/(e*m) - b*n*((e*log(f) + e *log(x^m) + d)*log((e*log(f) + e*log(x^m) + d)/e)/e - (e*log(f) + e*log(x^ m) + d)/e)/(e*m^2) + a*log((e*log(f) + e*log(x^m) + d)/e)/(e*m)
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx=\frac {b n \log \left (x\right )}{e m} + \frac {{\left (b e m \log \left (c\right ) - b e n \log \left (f\right ) + a e m - b d n\right )} \log \left (\frac {1}{4} \, {\left (\pi e m {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi e {\left (\mathrm {sgn}\left (f\right ) - 1\right )}\right )}^{2} + {\left (e m \log \left ({\left | x \right |}\right ) + e \log \left ({\left | f \right |}\right ) + d\right )}^{2}\right )}{2 \, e^{2} m^{2}} \]
b*n*log(x)/(e*m) + 1/2*(b*e*m*log(c) - b*e*n*log(f) + a*e*m - b*d*n)*log(1 /4*(pi*e*m*(sgn(x) - 1) + pi*e*(sgn(f) - 1))^2 + (e*m*log(abs(x)) + e*log( abs(f)) + d)^2)/(e^2*m^2)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+e\,\ln \left (f\,x^m\right )\right )} \,d x \]