Integrand size = 16, antiderivative size = 44 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \] Output:
ln(-e*x^n/d)*ln(c*(d+e*x^n)^p)/n+p*polylog(2,1+e*x^n/d)/n
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {d+e x^n}{d}\right )}{n} \] Input:
Integrate[Log[c*(d + e*x^n)^p]/x,x]
Output:
(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p] + p*PolyLog[2, (d + e*x^n)/d])/n
Time = 0.41 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2904, 2841, 2752}
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.
\(\displaystyle \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {\int x^{-n} \log \left (c \left (e x^n+d\right )^p\right )dx^n}{n}\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )-e p \int \frac {\log \left (-\frac {e x^n}{d}\right )}{e x^n+d}dx^n}{n}\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}\) |
Input:
Int[Log[c*(d + e*x^n)^p]/x,x]
Output:
(Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p] + p*PolyLog[2, 1 + (e*x^n)/d])/n
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 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 _.), 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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.66 (sec) , antiderivative size = 170, normalized size of antiderivative = 3.86
method | result | size |
risch | \(\ln \left (x \right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (x \right )-\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-p \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(170\) |
Input:
int(ln(c*(d+e*x^n)^p)/x,x,method=_RETURNVERBOSE)
Output:
ln(x)*ln((d+e*x^n)^p)+(1/2*I*Pi*csgn(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)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)-1/2*I*Pi*cs gn(I*c*(d+e*x^n)^p)^3+1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)+ln(c))*ln (x)-p/n*dilog((d+e*x^n)/d)-p*ln(x)*ln((d+e*x^n)/d)
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.36 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {n p \log \left (e x^{n} + d\right ) \log \left (x\right ) - n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + n \log \left (c\right ) \log \left (x\right ) - p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right )}{n} \] Input:
integrate(log(c*(d+e*x^n)^p)/x,x, algorithm="fricas")
Output:
(n*p*log(e*x^n + d)*log(x) - n*p*log(x)*log((e*x^n + d)/d) + n*log(c)*log( x) - p*dilog(-(e*x^n + d)/d + 1))/n
\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \] Input:
integrate(ln(c*(d+e*x**n)**p)/x,x)
Output:
Integral(log(c*(d + e*x**n)**p)/x, x)
\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \] Input:
integrate(log(c*(d+e*x^n)^p)/x,x, algorithm="maxima")
Output:
d*n*p*integrate(log(x)/(e*x*x^n + d*x), x) - 1/2*n*p*log(x)^2 + log((e*x^n + d)^p)*log(x) + log(c)*log(x)
\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x} \,d x } \] Input:
integrate(log(c*(d+e*x^n)^p)/x,x, algorithm="giac")
Output:
integrate(log((e*x^n + d)^p*c)/x, x)
Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x} \,d x \] Input:
int(log(c*(d + e*x^n)^p)/x,x)
Output:
int(log(c*(d + e*x^n)^p)/x, x)
\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}{x^{n} e x +d x}d x \right ) d n p +{\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right )}^{2}}{2 n p} \] Input:
int(log(c*(d+e*x^n)^p)/x,x)
Output:
(2*int(log((x**n*e + d)**p*c)/(x**n*e*x + d*x),x)*d*n*p + log((x**n*e + d) **p*c)**2)/(2*n*p)