Integrand size = 26, antiderivative size = 114 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^k\right )}{2 b n}-\frac {k \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^m}{e}\right )}{2 b n}-\frac {k \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right )}{m}+\frac {b k n \operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right )}{m^2} \]
1/2*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^m)^k)/b/n-1/2*k*(a+b*ln(c*x^n))^2*ln(1+f *x^m/e)/b/n-k*(a+b*ln(c*x^n))*polylog(2,-f*x^m/e)/m+b*k*n*polylog(3,-f*x^m /e)/m^2
Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(114)=228\).
Time = 0.14 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx=-\frac {1}{6} b k m n \log ^3(x)-\frac {1}{2} b k n \log ^2(x) \log \left (1+\frac {e x^{-m}}{f}\right )+b k n \log ^2(x) \log \left (e+f x^m\right )-\frac {b k n \log (x) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-b k \log (x) \log \left (c x^n\right ) \log \left (e+f x^m\right )+\frac {b k \log \left (-\frac {f x^m}{e}\right ) \log \left (c x^n\right ) \log \left (e+f x^m\right )}{m}-\frac {1}{2} b n \log ^2(x) \log \left (d \left (e+f x^m\right )^k\right )+\frac {a \log \left (-\frac {f x^m}{e}\right ) \log \left (d \left (e+f x^m\right )^k\right )}{m}+b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+\frac {b k n \log (x) \operatorname {PolyLog}\left (2,-\frac {e x^{-m}}{f}\right )}{m}+\frac {k \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {f x^m}{e}\right )}{m}+\frac {b k n \operatorname {PolyLog}\left (3,-\frac {e x^{-m}}{f}\right )}{m^2} \]
-1/6*(b*k*m*n*Log[x]^3) - (b*k*n*Log[x]^2*Log[1 + e/(f*x^m)])/2 + b*k*n*Lo g[x]^2*Log[e + f*x^m] - (b*k*n*Log[x]*Log[-((f*x^m)/e)]*Log[e + f*x^m])/m - b*k*Log[x]*Log[c*x^n]*Log[e + f*x^m] + (b*k*Log[-((f*x^m)/e)]*Log[c*x^n] *Log[e + f*x^m])/m - (b*n*Log[x]^2*Log[d*(e + f*x^m)^k])/2 + (a*Log[-((f*x ^m)/e)]*Log[d*(e + f*x^m)^k])/m + b*Log[x]*Log[c*x^n]*Log[d*(e + f*x^m)^k] + (b*k*n*Log[x]*PolyLog[2, -(e/(f*x^m))])/m + (k*(a - b*n*Log[x] + b*Log[ c*x^n])*PolyLog[2, 1 + (f*x^m)/e])/m + (b*k*n*PolyLog[3, -(e/(f*x^m))])/m^ 2
Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2822, 2775, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2822 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^k\right )}{2 b n}-\frac {f k m \int \frac {x^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{f x^m+e}dx}{2 b n}\) |
| \(\Big \downarrow \) 2775 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^k\right )}{2 b n}-\frac {f k m \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f m}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {f x^m}{e}+1\right )}{x}dx}{f m}\right )}{2 b n}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^k\right )}{2 b n}-\frac {f k m \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f m}-\frac {2 b n \left (\frac {b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right )}{x}dx}{m}-\frac {\operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}\right )}{f m}\right )}{2 b n}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^k\right )}{2 b n}-\frac {f k m \left (\frac {\log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f m}-\frac {2 b n \left (\frac {b n \operatorname {PolyLog}\left (3,-\frac {f x^m}{e}\right )}{m^2}-\frac {\operatorname {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}\right )}{f m}\right )}{2 b n}\) |
((a + b*Log[c*x^n])^2*Log[d*(e + f*x^m)^k])/(2*b*n) - (f*k*m*(((a + b*Log[ c*x^n])^2*Log[1 + (f*x^m)/e])/(f*m) - (2*b*n*(-(((a + b*Log[c*x^n])*PolyLo g[2, -((f*x^m)/e)])/m) + (b*n*PolyLog[3, -((f*x^m)/e)])/m^2))/(f*m)))/(2*b *n)
3.2.47.3.1 Defintions of rubi rules used
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Simp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c* x^n])^p/(e*r)), x] - Simp[b*f^m*n*(p/(e*r)) Int[Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] & & EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]
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[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_ .)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp[Log[d*(e + f*x^m)^r]*((a + b*Log[ c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[f*m*(r/(b*n*(p + 1))) Int[x^(m - 1)*((a + b*Log[c*x^n])^(p + 1)/(e + f*x^m)), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && NeQ[d*e, 1]
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]
\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )}{x}d x\]
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx=\frac {b m^{2} n \log \left (d\right ) \log \left (x\right )^{2} + 2 \, b k n {\rm polylog}\left (3, -\frac {f x^{m}}{e}\right ) + 2 \, {\left (b m^{2} \log \left (c\right ) + a m^{2}\right )} \log \left (d\right ) \log \left (x\right ) - 2 \, {\left (b k m n \log \left (x\right ) + b k m \log \left (c\right ) + a k m\right )} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) + {\left (b k m^{2} n \log \left (x\right )^{2} + 2 \, {\left (b k m^{2} \log \left (c\right ) + a k m^{2}\right )} \log \left (x\right )\right )} \log \left (f x^{m} + e\right ) - {\left (b k m^{2} n \log \left (x\right )^{2} + 2 \, {\left (b k m^{2} \log \left (c\right ) + a k m^{2}\right )} \log \left (x\right )\right )} \log \left (\frac {f x^{m} + e}{e}\right )}{2 \, m^{2}} \]
1/2*(b*m^2*n*log(d)*log(x)^2 + 2*b*k*n*polylog(3, -f*x^m/e) + 2*(b*m^2*log (c) + a*m^2)*log(d)*log(x) - 2*(b*k*m*n*log(x) + b*k*m*log(c) + a*k*m)*dil og(-(f*x^m + e)/e + 1) + (b*k*m^2*n*log(x)^2 + 2*(b*k*m^2*log(c) + a*k*m^2 )*log(x))*log(f*x^m + e) - (b*k*m^2*n*log(x)^2 + 2*(b*k*m^2*log(c) + a*k*m ^2)*log(x))*log((f*x^m + e)/e))/m^2
Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx=\text {Exception raised: TypeError} \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{m} + e\right )}^{k} d\right )}{x} \,d x } \]
-1/2*(b*n*log(x)^2 - 2*b*log(x)*log(x^n) - 2*(b*log(c) + a)*log(x))*log((f *x^m + e)^k) - integrate(-1/2*(2*b*e*log(c)*log(d) + 2*a*e*log(d) + (b*f*k *m*n*log(x)^2 + 2*b*f*log(c)*log(d) + 2*a*f*log(d) - 2*(b*f*k*m*log(c) + a *f*k*m)*log(x))*x^m + 2*(b*e*log(d) - (b*f*k*m*log(x) - b*f*log(d))*x^m)*l og(x^n))/(f*x*x^m + e*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{m} + e\right )}^{k} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]