Integrand size = 24, antiderivative size = 164 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx=\frac {b f m n \log (x)}{e}-\frac {b f m n \log ^2(x)}{2 e}+\frac {f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b f m n \log (e+f x)}{e}+\frac {b f m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{e}-\frac {f m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{e}-\frac {b n \log \left (d (e+f x)^m\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}+\frac {b f m n \operatorname {PolyLog}\left (2,1+\frac {f x}{e}\right )}{e} \]
b*f*m*n*ln(x)/e-1/2*b*f*m*n*ln(x)^2/e+f*m*ln(x)*(a+b*ln(c*x^n))/e-b*f*m*n* ln(f*x+e)/e+b*f*m*n*ln(-f*x/e)*ln(f*x+e)/e-f*m*(a+b*ln(c*x^n))*ln(f*x+e)/e -b*n*ln(d*(f*x+e)^m)/x-(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)/x+b*f*m*n*polylog(2 ,1+f*x/e)/e
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx=-\frac {b f m n x \log ^2(x)+2 \left (a+b n+b \log \left (c x^n\right )\right ) \left (f m x \log (e+f x)+e \log \left (d (e+f x)^m\right )\right )-2 f m x \log (x) \left (a+b n+b \log \left (c x^n\right )+b n \log (e+f x)-b n \log \left (1+\frac {f x}{e}\right )\right )+2 b f m n x \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{2 e x} \]
-1/2*(b*f*m*n*x*Log[x]^2 + 2*(a + b*n + b*Log[c*x^n])*(f*m*x*Log[e + f*x] + e*Log[d*(e + f*x)^m]) - 2*f*m*x*Log[x]*(a + b*n + b*Log[c*x^n] + b*n*Log [e + f*x] - b*n*Log[1 + (f*x)/e]) + 2*b*f*m*n*x*PolyLog[2, -((f*x)/e)])/(e *x)
Time = 0.33 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2823, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (\frac {f m \log (x)}{e x}-\frac {f m \log (e+f x)}{e x}-\frac {\log \left (d (e+f x)^m\right )}{x^2}\right )dx-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}+\frac {f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {f m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x}+\frac {f m \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {f m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{e}-b n \left (\frac {\log \left (d (e+f x)^m\right )}{x}-\frac {f m \operatorname {PolyLog}\left (2,\frac {f x}{e}+1\right )}{e}+\frac {f m \log ^2(x)}{2 e}-\frac {f m \log (x)}{e}+\frac {f m \log (e+f x)}{e}-\frac {f m \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{e}\right )\) |
(f*m*Log[x]*(a + b*Log[c*x^n]))/e - (f*m*(a + b*Log[c*x^n])*Log[e + f*x])/ e - ((a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/x - b*n*(-((f*m*Log[x])/e) + ( f*m*Log[x]^2)/(2*e) + (f*m*Log[e + f*x])/e - (f*m*Log[-((f*x)/e)]*Log[e + f*x])/e + Log[d*(e + f*x)^m]/x - (f*m*PolyLog[2, 1 + (f*x)/e])/e)
3.1.75.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.28 (sec) , antiderivative size = 737, normalized size of antiderivative = 4.49
| method | result | size |
| risch | \(\left (-\frac {b \ln \left (x^{n}\right )}{x}-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b n +2 a}{2 x}\right ) \ln \left (\left (f x +e \right )^{m}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (f x +e \right )^{m}\right ) \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right )^{2}}{4}-\frac {i \pi \,\operatorname {csgn}\left (i \left (f x +e \right )^{m}\right ) \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right ) \operatorname {csgn}\left (i d \right )}{4}-\frac {i \pi \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right )^{3}}{4}+\frac {i \pi \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right )^{2} \operatorname {csgn}\left (i d \right )}{4}+\frac {\ln \left (d \right )}{2}\right ) \left (-\frac {-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a}{x}-\frac {2 b \ln \left (x^{n}\right )}{x}-\frac {2 b n}{x}\right )+\frac {i m f \ln \left (f x +e \right ) b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2 e}-\frac {i m f \ln \left (f x +e \right ) b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2 e}+\frac {i m f \ln \left (x \right ) b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2 e}-\frac {i m f \ln \left (x \right ) b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2 e}+\frac {m f \ln \left (x \right ) b \ln \left (c \right )}{e}+\frac {b f m n \ln \left (x \right )}{e}+\frac {m f \ln \left (x \right ) a}{e}+\frac {i m f \ln \left (f x +e \right ) b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2 e}+\frac {i m f \ln \left (x \right ) b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2 e}-\frac {i m f \ln \left (f x +e \right ) b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2 e}-\frac {i m f \ln \left (x \right ) b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2 e}-\frac {m f \ln \left (f x +e \right ) b \ln \left (c \right )}{e}-\frac {b f m n \ln \left (f x +e \right )}{e}-\frac {m f \ln \left (f x +e \right ) a}{e}+\frac {m f b \ln \left (x^{n}\right ) \ln \left (x \right )}{e}-\frac {m f b \ln \left (x^{n}\right ) \ln \left (f x +e \right )}{e}-\frac {b f m n \ln \left (x \right )^{2}}{2 e}+\frac {b f m n \ln \left (-\frac {f x}{e}\right ) \ln \left (f x +e \right )}{e}+\frac {m f b n \operatorname {dilog}\left (-\frac {f x}{e}\right )}{e}\) | \(737\) |
(-b/x*ln(x^n)-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*b*n+2*a)/x)*ln((f*x+e)^m)+(1/4*I*Pi*csgn(I*(f*x+e)^m)*cs gn(I*d*(f*x+e)^m)^2-1/4*I*Pi*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*csgn(I* d)-1/4*I*Pi*csgn(I*d*(f*x+e)^m)^3+1/4*I*Pi*csgn(I*d*(f*x+e)^m)^2*csgn(I*d) +1/2*ln(d))*(-(-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)/x-2*b/x*ln(x^n)-2*b*n/x)+1/2*I*m*f/e*ln(f*x+e)*b*Pi*csgn( I*c*x^n)^3-1/2*I*m*f/e*ln(f*x+e)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*m*f/ e*ln(x)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m*f/e*ln(x)*b*Pi*csgn(I*c*x^n )^3+m*f/e*ln(x)*b*ln(c)+b*f*m*n*ln(x)/e+m*f/e*ln(x)*a+1/2*I*m*f/e*ln(f*x+e )*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*m*f/e*ln(x)*b*Pi*csgn(I*x ^n)*csgn(I*c*x^n)^2-1/2*I*m*f/e*ln(f*x+e)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2 -1/2*I*m*f/e*ln(x)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-m*f/e*ln(f*x+e )*b*ln(c)-b*f*m*n*ln(f*x+e)/e-m*f/e*ln(f*x+e)*a+m*f*b*ln(x^n)/e*ln(x)-m*f* b*ln(x^n)/e*ln(f*x+e)-1/2*b*f*m*n*ln(x)^2/e+b*f*m*n*ln(-f*x/e)*ln(f*x+e)/e +m*f*b*n/e*dilog(-f*x/e)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx=-\frac {{\left (\log \left (\frac {f x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {f x}{e}\right )\right )} b f m n}{e} - \frac {{\left (a f m + {\left (f m n + f m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{e} + \frac {2 \, b f m n x \log \left (f x + e\right ) \log \left (x\right ) - b f m n x \log \left (x\right )^{2} - 2 \, a e \log \left (d\right ) + 2 \, {\left (a f m + {\left (f m n + f m \log \left (c\right )\right )} b\right )} x \log \left (x\right ) - 2 \, {\left (e n \log \left (d\right ) + e \log \left (c\right ) \log \left (d\right )\right )} b - 2 \, {\left (b e \log \left (x^{n}\right ) + {\left (e n + e \log \left (c\right )\right )} b + a e\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 2 \, {\left (b f m x \log \left (f x + e\right ) - b f m x \log \left (x\right ) + b e \log \left (d\right )\right )} \log \left (x^{n}\right )}{2 \, e x} \]
-(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*f*m*n/e - (a*f*m + (f*m*n + f*m *log(c))*b)*log(f*x + e)/e + 1/2*(2*b*f*m*n*x*log(f*x + e)*log(x) - b*f*m* n*x*log(x)^2 - 2*a*e*log(d) + 2*(a*f*m + (f*m*n + f*m*log(c))*b)*x*log(x) - 2*(e*n*log(d) + e*log(c)*log(d))*b - 2*(b*e*log(x^n) + (e*n + e*log(c))* b + a*e)*log((f*x + e)^m) - 2*(b*f*m*x*log(f*x + e) - b*f*m*x*log(x) + b*e *log(d))*log(x^n))/(e*x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^2} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]