Integrand size = 18, antiderivative size = 113 \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac {3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n}-\frac {6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (3,1+\frac {e x^n}{d}\right )}{n}+\frac {6 p^3 \operatorname {PolyLog}\left (4,1+\frac {e x^n}{d}\right )}{n} \]
ln(-e*x^n/d)*ln(c*(d+e*x^n)^p)^3/n+3*p*ln(c*(d+e*x^n)^p)^2*polylog(2,1+e*x ^n/d)/n-6*p^2*ln(c*(d+e*x^n)^p)*polylog(3,1+e*x^n/d)/n+6*p^3*polylog(4,1+e *x^n/d)/n
Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(113)=226\).
Time = 0.11 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.39 \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {-n p^3 \log (x) \log ^3\left (d+e x^n\right )+p^3 \log \left (-\frac {e x^n}{d}\right ) \log ^3\left (d+e x^n\right )+3 n p^2 \log (x) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 p^2 \log \left (-\frac {e x^n}{d}\right ) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 n p \log (x) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+3 p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+n \log (x) \log ^3\left (c \left (d+e x^n\right )^p\right )+3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )-6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \operatorname {PolyLog}\left (3,1+\frac {e x^n}{d}\right )+6 p^3 \operatorname {PolyLog}\left (4,1+\frac {e x^n}{d}\right )}{n} \]
(-(n*p^3*Log[x]*Log[d + e*x^n]^3) + p^3*Log[-((e*x^n)/d)]*Log[d + e*x^n]^3 + 3*n*p^2*Log[x]*Log[d + e*x^n]^2*Log[c*(d + e*x^n)^p] - 3*p^2*Log[-((e*x ^n)/d)]*Log[d + e*x^n]^2*Log[c*(d + e*x^n)^p] - 3*n*p*Log[x]*Log[d + e*x^n ]*Log[c*(d + e*x^n)^p]^2 + 3*p*Log[-((e*x^n)/d)]*Log[d + e*x^n]*Log[c*(d + e*x^n)^p]^2 + n*Log[x]*Log[c*(d + e*x^n)^p]^3 + 3*p*Log[c*(d + e*x^n)^p]^ 2*PolyLog[2, 1 + (e*x^n)/d] - 6*p^2*Log[c*(d + e*x^n)^p]*PolyLog[3, 1 + (e *x^n)/d] + 6*p^3*PolyLog[4, 1 + (e*x^n)/d])/n
Time = 0.48 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2904, 2843, 2881, 2821, 2830, 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.
\(\displaystyle \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {\int x^{-n} \log ^3\left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
\(\Big \downarrow \) 2843 |
\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 e p \int \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (e x^n+d\right )^p\right )}{e x^n+d}dx^n}{n}\) |
\(\Big \downarrow \) 2881 |
\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \int x^{-n} \log \left (-\frac {e x^n}{d}\right ) \log ^2\left (c \left (e x^n+d\right )^p\right )d\left (e x^n+d\right )}{n}\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \left (2 p \int x^{-n} \log \left (c \left (e x^n+d\right )^p\right ) \operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right )d\left (e x^n+d\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 2830 |
\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \left (2 p \left (\operatorname {PolyLog}\left (3,\frac {e x^n+d}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-p \int x^{-n} \operatorname {PolyLog}\left (3,\frac {e x^n+d}{d}\right )d\left (e x^n+d\right )\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )-3 p \left (2 p \left (\operatorname {PolyLog}\left (3,\frac {e x^n+d}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-p \operatorname {PolyLog}\left (4,\frac {e x^n+d}{d}\right )\right )-\operatorname {PolyLog}\left (2,\frac {e x^n+d}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )\right )}{n}\) |
(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p]^3 - 3*p*(-(Log[c*(d + e*x^n)^p]^2* PolyLog[2, (d + e*x^n)/d]) + 2*p*(Log[c*(d + e*x^n)^p]*PolyLog[3, (d + e*x ^n)/d] - p*PolyLog[4, (d + e*x^n)/d])))/n
3.2.75.3.1 Defintions of rubi rules used
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[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[b*n*(p/q) Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Simp[b*e*n*(p/g) Int[Log[(e*(f + g*x))/(e*f - d*g)] *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym bol] :> Simp[1/e Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* ((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
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 = 2.99 (sec) , antiderivative size = 1409, normalized size of antiderivative = 12.47
-1/n*ln(e*x^n)*ln(d+e*x^n)^3*p^3+3/n*ln(-e*x^n/d)*ln(d+e*x^n)^3*p^3-2/n*ln (1-(d+e*x^n)/d)*ln(d+e*x^n)^3*p^3+3/n*ln(e*x^n)*ln((d+e*x^n)^p)*ln(d+e*x^n )^2*p^2+3/n*dilog(-e*x^n/d)*ln(d+e*x^n)^2*p^3-6/n*ln(-e*x^n/d)*ln((d+e*x^n )^p)*ln(d+e*x^n)^2*p^2+3/n*ln(1-(d+e*x^n)/d)*ln((d+e*x^n)^p)*ln(d+e*x^n)^2 *p^2-3/n*polylog(2,(d+e*x^n)/d)*ln(d+e*x^n)^2*p^3-3/n*ln(e*x^n)*ln((d+e*x^ n)^p)^2*ln(d+e*x^n)*p-6/n*dilog(-e*x^n/d)*ln((d+e*x^n)^p)*ln(d+e*x^n)*p^2+ 3/n*ln(-e*x^n/d)*ln((d+e*x^n)^p)^2*ln(d+e*x^n)*p+6/n*polylog(2,(d+e*x^n)/d )*ln((d+e*x^n)^p)*ln(d+e*x^n)*p^2+1/n*ln(e*x^n)*ln((d+e*x^n)^p)^3+3/n*dilo g(-e*x^n/d)*ln((d+e*x^n)^p)^2*p-6/n*polylog(3,(d+e*x^n)/d)*ln((d+e*x^n)^p) *p^2+6/n*polylog(4,(d+e*x^n)/d)*p^3+1/8*(I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c *(d+e*x^n)^p)^2-I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-I *Pi*csgn(I*c*(d+e*x^n)^p)^3+I*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+2*ln(c) )^3*ln(x)+(3/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2-3/2*I*Pi*c sgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-3/2*I*Pi*csgn(I*c*(d+e* x^n)^p)^3+3/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+3*ln(c))/n*((ln((d+e* x^n)^p)-p*ln(d+e*x^n))^2*ln(e*x^n)+p^2*(ln(d+e*x^n)^2*ln(1-(d+e*x^n)/d)+2* ln(d+e*x^n)*polylog(2,(d+e*x^n)/d)-2*polylog(3,(d+e*x^n)/d))+2*p*(ln((d+e* x^n)^p)-p*ln(d+e*x^n))*(dilog(-e*x^n/d)+ln(d+e*x^n)*ln(-e*x^n/d)))+(-3/4*P i^2*csgn(I*(d+e*x^n)^p)^2*csgn(I*c*(d+e*x^n)^p)^4+3/2*Pi^2*csgn(I*(d+e*x^n )^p)^2*csgn(I*c*(d+e*x^n)^p)^3*csgn(I*c)-3/4*Pi^2*csgn(I*(d+e*x^n)^p)^2...
\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{3}}{x} \,d x } \]
\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{3}}{x}\, dx \]
\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{3}}{x} \,d x } \]
log((e*x^n + d)^p)^3*log(x) - integrate(-(e*x^n*log(c)^3 + d*log(c)^3 - 3* ((e*n*p*log(x) - e*log(c))*x^n - d*log(c))*log((e*x^n + d)^p)^2 + 3*(e*x^n *log(c)^2 + d*log(c)^2)*log((e*x^n + d)^p))/(e*x*x^n + d*x), x)
\[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{3}}{x} \,d x } \]
Timed out. \[ \int \frac {\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^3}{x} \,d x \]