Integrand size = 27, antiderivative size = 70 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n} \]
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (g+f x^n\right )}{-d f+e g}\right )+p \operatorname {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n} \]
(Log[c*(d + e*x^n)^p]*Log[(e*(g + f*x^n))/(-(d*f) + e*g)] + p*PolyLog[2, ( f*(d + e*x^n))/(d*f - e*g)])/(f*n)
Time = 0.37 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2005, 2925, 2841, 2840, 2838}
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.
\(\displaystyle \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx\) |
\(\Big \downarrow \) 2005 |
\(\displaystyle \int \frac {x^{n-1} \log \left (c \left (d+e x^n\right )^p\right )}{f x^n+g}dx\) |
\(\Big \downarrow \) 2925 |
\(\displaystyle \frac {\int \frac {\log \left (c \left (e x^n+d\right )^p\right )}{f x^n+g}dx^n}{n}\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle \frac {\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f}-\frac {e p \int \frac {\log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{e x^n+d}dx^n}{f}}{n}\) |
\(\Big \downarrow \) 2840 |
\(\displaystyle \frac {\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f}-\frac {p \int x^{-n} \log \left (1-\frac {f \left (e x^n+d\right )}{d f-e g}\right )d\left (e x^n+d\right )}{f}}{n}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f}+\frac {p \operatorname {PolyLog}\left (2,\frac {f \left (e x^n+d\right )}{d f-e g}\right )}{f}}{n}\) |
((Log[c*(d + e*x^n)^p]*Log[-((e*(g + f*x^n))/(d*f - e*g))])/f + (p*PolyLog [2, (f*(d + e*x^n))/(d*f - e*g)])/f)/n
3.4.72.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ Symbol] :> Simp[1/g Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c *(e*f - d*g), 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ )), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x )^n])/g), x] - Simp[b*e*(n/g) Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.47
method | result | size |
risch | \(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (g +f \,x^{n}\right )}{n f}-\frac {p \operatorname {dilog}\left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n f}-\frac {p \ln \left (g +f \,x^{n}\right ) \ln \left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n f}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (g +f \,x^{n}\right )}{n f}\) | \(243\) |
1/n*ln((d+e*x^n)^p)*ln(g+f*x^n)/f-1/n/f*p*dilog(((g+f*x^n)*e+d*f-e*g)/(d*f -e*g))-1/n/f*p*ln(g+f*x^n)*ln(((g+f*x^n)*e+d*f-e*g)/(d*f-e*g))+(1/2*I*Pi*c sgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2-1/2*I*Pi*csgn(I*(d+e*x^n)^p)*cs gn(I*c*(d+e*x^n)^p)*csgn(I*c)-1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3+1/2*I*Pi*cs gn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+ln(c))/n*ln(g+f*x^n)/f
\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{n}}\right )} x} \,d x } \]
Exception generated. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.60 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx={\left (\frac {\log \left (f + \frac {g}{x^{n}}\right )}{f n} - \frac {\log \left (\frac {1}{x^{n}}\right )}{f n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) - \frac {{\left (\log \left (f x^{n} + g\right ) \log \left (\frac {e f x^{n} + e g}{d f - e g} + 1\right ) + {\rm Li}_2\left (-\frac {e f x^{n} + e g}{d f - e g}\right )\right )} p}{f n} \]
(log(f + g/x^n)/(f*n) - log(1/(x^n))/(f*n))*log((e*x^n + d)^p*c) - (log(f* x^n + g)*log((e*f*x^n + e*g)/(d*f - e*g) + 1) + dilog(-(e*f*x^n + e*g)/(d* f - e*g)))*p/(f*n)
\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{n}}\right )} x} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,\left (f+\frac {g}{x^n}\right )} \,d x \]