Integrand size = 21, antiderivative size = 117 \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=2 b m n x-m x \left (a+b \log \left (c x^n\right )\right )-\frac {b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac {b e n \log \left (-\frac {f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+\frac {(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-\frac {b e m n \operatorname {PolyLog}\left (2,1+\frac {f x}{e}\right )}{f} \]
2*b*m*n*x-m*x*(a+b*ln(c*x^n))-b*n*(f*x+e)*ln(d*(f*x+e)^m)/f-b*e*n*ln(-f*x/ e)*ln(d*(f*x+e)^m)/f+(f*x+e)*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)/f-b*e*m*n*pol ylog(2,1+f*x/e)/f
Time = 0.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.30 \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\frac {-a f m x+2 b f m n x-b e m n \log (e+f x)-b e m n \log (x) \log (e+f x)+a e \log \left (d (e+f x)^m\right )+a f x \log \left (d (e+f x)^m\right )-b f n x \log \left (d (e+f x)^m\right )+b \log \left (c x^n\right ) \left (e m \log (e+f x)+f x \left (-m+\log \left (d (e+f x)^m\right )\right )\right )+b e m n \log (x) \log \left (1+\frac {f x}{e}\right )+b e m n \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{f} \]
(-(a*f*m*x) + 2*b*f*m*n*x - b*e*m*n*Log[e + f*x] - b*e*m*n*Log[x]*Log[e + f*x] + a*e*Log[d*(e + f*x)^m] + a*f*x*Log[d*(e + f*x)^m] - b*f*n*x*Log[d*( e + f*x)^m] + b*Log[c*x^n]*(e*m*Log[e + f*x] + f*x*(-m + Log[d*(e + f*x)^m ])) + b*e*m*n*Log[x]*Log[1 + (f*x)/e] + b*e*m*n*PolyLog[2, -((f*x)/e)])/f
Time = 0.35 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2817, 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.
| \(\displaystyle \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx\) |
| \(\Big \downarrow \) 2817 |
| \(\displaystyle -b n \int \left (\frac {(e+f x) \log \left (d (e+f x)^m\right )}{f x}-m\right )dx+\frac {(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-m x \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-m x \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {(e+f x) \log \left (d (e+f x)^m\right )}{f}+\frac {e \log \left (-\frac {f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+\frac {e m \operatorname {PolyLog}\left (2,\frac {f x}{e}+1\right )}{f}-2 m x\right )\) |
-(m*x*(a + b*Log[c*x^n])) + ((e + f*x)*(a + b*Log[c*x^n])*Log[d*(e + f*x)^ m])/f - b*n*(-2*m*x + ((e + f*x)*Log[d*(e + f*x)^m])/f + (e*Log[-((f*x)/e) ]*Log[d*(e + f*x)^m])/f + (e*m*PolyLog[2, 1 + (f*x)/e])/f)
3.1.73.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n])^p u, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1)/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.71 (sec) , antiderivative size = 686, normalized size of antiderivative = 5.86
| method | result | size |
| risch | \(\left (b x \ln \left (x^{n}\right )+\frac {x \left (-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 \right )}{2}\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 (i \pi b x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 a x +2 \ln \left (c \right ) b x +2 b x \ln \left (x^{n}\right )-2 b n x -i \pi b x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )\right )-\frac {i m x b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i m x b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}-\frac {i m x b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i m e \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 f}-m x b \ln \left (c \right )+2 b m n x -x a m +\frac {i m x b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i m e \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 f}+\frac {i m e \ln \left (f x +e \right ) b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2 f}-\frac {i m e \ln \left (f x +e \right ) b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2 f}+\frac {m e \ln \left (f x +e \right ) b \ln \left (c \right )}{f}-\frac {m b n e \ln \left (f x +e \right )}{f}+\frac {a m e \ln \left (f x +e \right )}{f}-m b \ln \left (x^{n}\right ) x +\frac {m b \ln \left (x^{n}\right ) e \ln \left (f x +e \right )}{f}+\frac {m b n e}{f}-\frac {m b n e \ln \left (f x +e \right ) \ln \left (-\frac {f x}{e}\right )}{f}-\frac {m b n e \operatorname {dilog}\left (-\frac {f x}{e}\right )}{f}\) | \(686\) |
(b*x*ln(x^n)+1/2*x*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csg n(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))*ln((f*x+e)^m)+(1/4*I*Pi*csgn(I*(f*x+e)^m)*csg n(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*Pi*b*x*csgn(I*c)*csgn(I*c*x^n)^2+I*Pi*b*x*csgn(I*x^n)*csgn(I *c*x^n)^2+2*a*x+2*ln(c)*b*x+2*b*x*ln(x^n)-2*b*n*x-I*Pi*b*x*csgn(I*c*x^n)^3 -I*Pi*b*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n))-1/2*I*m*x*b*Pi*csgn(I*x^n)* csgn(I*c*x^n)^2+1/2*I*m*x*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*m *x*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m/f*e*ln(f*x+e)*b*Pi*csgn(I*c)*csg n(I*x^n)*csgn(I*c*x^n)-m*x*b*ln(c)+2*b*m*n*x-x*a*m+1/2*I*m*x*b*Pi*csgn(I*c *x^n)^3+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(f*x+e)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m/f*e*ln(f*x+e)*b*Pi*csgn( I*c*x^n)^3+m/f*e*ln(f*x+e)*b*ln(c)-m/f*b*n*e*ln(f*x+e)+a*m/f*e*ln(f*x+e)-m *b*ln(x^n)*x+m/f*b*ln(x^n)*e*ln(f*x+e)+m/f*b*n*e-m/f*b*n*e*ln(f*x+e)*ln(-f *x/e)-m/f*b*n*e*dilog(-f*x/e)
\[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\text {Timed out} \]
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.61 \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\frac {{\left (\log \left (\frac {f x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {f x}{e}\right )\right )} b e m n}{f} + \frac {{\left (a e m - {\left (e m n - e m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{f} - \frac {b e m n \log \left (f x + e\right ) \log \left (x\right ) + {\left ({\left (f m - f \log \left (d\right )\right )} a - {\left (2 \, f m n - f n \log \left (d\right ) - {\left (f m - f \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x - {\left (b f x \log \left (x^{n}\right ) - {\left ({\left (f n - f \log \left (c\right )\right )} b - a f\right )} x\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - {\left (b e m \log \left (f x + e\right ) - {\left (f m - f \log \left (d\right )\right )} b x\right )} \log \left (x^{n}\right )}{f} \]
(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*e*m*n/f + (a*e*m - (e*m*n - e*m* log(c))*b)*log(f*x + e)/f - (b*e*m*n*log(f*x + e)*log(x) + ((f*m - f*log(d ))*a - (2*f*m*n - f*n*log(d) - (f*m - f*log(d))*log(c))*b)*x - (b*f*x*log( x^n) - ((f*n - f*log(c))*b - a*f)*x)*log((f*x + e)^m) - (b*e*m*log(f*x + e ) - (f*m - f*log(d))*b*x)*log(x^n))/f
\[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\int \ln \left (d\,{\left (e+f\,x\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]