Integrand size = 24, antiderivative size = 88 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b n \log ^2\left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 m}-b n \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+b m n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right ) \]
1/2*ln(f*x^m)^2*(a+b*ln(c*(e*x+d)^n))/m-1/2*b*n*ln(f*x^m)^2*ln(1+e*x/d)/m- b*n*ln(f*x^m)*polylog(2,-e*x/d)+b*m*n*polylog(3,-e*x/d)
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.45 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\frac {1}{2} \left (\frac {a \log ^2\left (f x^m\right )}{m}-b m \log ^2(x) \log \left (c (d+e x)^n\right )+2 b \log (x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )+b m n \log ^2(x) \log \left (1+\frac {e x}{d}\right )-2 b n \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )-2 b n \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+2 b m n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right ) \]
((a*Log[f*x^m]^2)/m - b*m*Log[x]^2*Log[c*(d + e*x)^n] + 2*b*Log[x]*Log[f*x ^m]*Log[c*(d + e*x)^n] + b*m*n*Log[x]^2*Log[1 + (e*x)/d] - 2*b*n*Log[x]*Lo g[f*x^m]*Log[1 + (e*x)/d] - 2*b*n*Log[f*x^m]*PolyLog[2, -((e*x)/d)] + 2*b* m*n*PolyLog[3, -((e*x)/d)])/2
Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2872, 2754, 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.
\(\displaystyle \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 2872 |
\(\displaystyle \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b e n \int \frac {\log ^2\left (f x^m\right )}{d+e x}dx}{2 m}\) |
\(\Big \downarrow \) 2754 |
\(\displaystyle \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b e n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \log ^2\left (f x^m\right )}{e}-\frac {2 m \int \frac {\log \left (f x^m\right ) \log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{2 m}\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b e n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \log ^2\left (f x^m\right )}{e}-\frac {2 m \left (m \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \log \left (f x^m\right )\right )}{e}\right )}{2 m}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b e n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \log ^2\left (f x^m\right )}{e}-\frac {2 m \left (m \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \log \left (f x^m\right )\right )}{e}\right )}{2 m}\) |
(Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m) - (b*e*n*((Log[f*x^m]^2*Lo g[1 + (e*x)/d])/e - (2*m*(-(Log[f*x^m]*PolyLog[2, -((e*x)/d)]) + m*PolyLog [3, -((e*x)/d)]))/e))/(2*m)
3.4.62.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
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]
Int[(Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b _.)))/(x_), x_Symbol] :> Simp[Log[f*x^m]^2*((a + b*Log[c*(d + e*x)^n])/(2*m )), x] - Simp[b*e*(n/(2*m)) Int[Log[f*x^m]^2/(d + e*x), x], x] /; FreeQ[{ a, b, c, d, e, f, m, n}, x]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.60 (sec) , antiderivative size = 756, normalized size of antiderivative = 8.59
(b*ln(x)*ln(x^m)-1/2*b*m*ln(x)^2-1/2*I*Pi*ln(x)*b*csgn(I*f)*csgn(I*x^m)*cs gn(I*f*x^m)+1/2*I*Pi*ln(x)*b*csgn(I*f)*csgn(I*f*x^m)^2+1/2*I*Pi*ln(x)*b*cs gn(I*x^m)*csgn(I*f*x^m)^2-1/2*I*Pi*ln(x)*b*csgn(I*f*x^m)^3+ln(f)*ln(x)*b)* ln((e*x+d)^n)+(-1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n) +1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*I*b*Pi*csgn(I*(e*x+d)^n)*c sgn(I*c*(e*x+d)^n)^2-1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3+1/2*b*ln(c)+1/2*a)*( I*Pi*csgn(I*f)*csgn(I*f*x^m)^2*ln(x)+I*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2*ln(x )+2*ln(x)*ln(f)-I*Pi*csgn(I*f*x^m)^3*ln(x)-I*Pi*csgn(I*f)*csgn(I*x^m)*csgn (I*f*x^m)*ln(x)+1/m*ln(x^m)^2)-1/2*I*n*b*ln(x)*ln((e*x+d)/d)*Pi*csgn(I*x^m )*csgn(I*f*x^m)^2-1/2*I*n*b*dilog((e*x+d)/d)*Pi*csgn(I*f)*csgn(I*f*x^m)^2- 1/2*I*n*b*dilog((e*x+d)/d)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/2*I*n*b*dilog( (e*x+d)/d)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/2*I*n*b*ln(x)*ln((e*x+ d)/d)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/2*I*n*b*ln(x)*ln((e*x+d)/d) *Pi*csgn(I*f)*csgn(I*f*x^m)^2+1/2*I*n*b*dilog((e*x+d)/d)*Pi*csgn(I*f*x^m)^ 3+1/2*I*n*b*ln(x)*ln((e*x+d)/d)*Pi*csgn(I*f*x^m)^3-1/2*n*b*m*ln(x)^2*ln(1+ e*x/d)+n*b*ln(x)^2*ln((e*x+d)/d)*m-n*b*m*ln(x)*polylog(2,-e*x/d)+n*b*dilog ((e*x+d)/d)*m*ln(x)-n*b*ln(x)*ln((e*x+d)/d)*ln(x^m)-n*b*ln(x)*ln((e*x+d)/d )*ln(f)+b*m*n*polylog(3,-e*x/d)-n*b*dilog((e*x+d)/d)*ln(x^m)-n*b*dilog((e* x+d)/d)*ln(f)
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x} \,d x } \]
-1/2*(b*m*log(x)^2 - 2*b*log(f)*log(x) - 2*b*log(x)*log(x^m))*log((e*x + d )^n) - integrate(-1/2*(b*e*m*n*x*log(x)^2 - 2*b*e*n*x*log(f)*log(x) + 2*b* d*log(c)*log(f) + 2*a*d*log(f) + 2*(b*e*log(c)*log(f) + a*e*log(f))*x - 2* (b*e*n*x*log(x) - b*d*log(c) - a*d - (b*e*log(c) + a*e)*x)*log(x^m))/(e*x^ 2 + d*x), x)
\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int \frac {\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x} \,d x \]