Integrand size = 20, antiderivative size = 49 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\frac {\log \left (-\frac {e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{m}+\frac {b n \operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )}{m} \] Output:
ln(-e*x^m/d)*(a+b*ln(c*(d+e*x^m)^n))/m+b*n*polylog(2,1+e*x^m/d)/m
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=a \log (x)+\frac {b \left (\log \left (-\frac {e x^m}{d}\right ) \log \left (c \left (d+e x^m\right )^n\right )+n \operatorname {PolyLog}\left (2,\frac {d+e x^m}{d}\right )\right )}{m} \] Input:
Integrate[(a + b*Log[c*(d + e*x^m)^n])/x,x]
Output:
a*Log[x] + (b*(Log[-((e*x^m)/d)]*Log[c*(d + e*x^m)^n] + n*PolyLog[2, (d + e*x^m)/d]))/m
Time = 0.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {\int x^{-m} \left (a+b \log \left (c \left (e x^m+d\right )^n\right )\right )dx^m}{m}\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle \frac {\log \left (-\frac {e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )-b e n \int \frac {\log \left (-\frac {e x^m}{d}\right )}{e x^m+d}dx^m}{m}\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {\log \left (-\frac {e x^m}{d}\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x^m}{d}+1\right )}{m}\) |
Input:
Int[(a + b*Log[c*(d + e*x^m)^n])/x,x]
Output:
(Log[-((e*x^m)/d)]*(a + b*Log[c*(d + e*x^m)^n]) + b*n*PolyLog[2, 1 + (e*x^ m)/d])/m
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 = 4.78 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.67
method | result | size |
risch | \(b \ln \left (x \right ) \ln \left (\left (d +e \,x^{m}\right )^{n}\right )+\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{m}\right )^{n}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right )}^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{m}\right )^{n}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right )}^{3}}{2}+\frac {i b \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{m}\right )^{n}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \ln \left (x \right )-\frac {b n \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right )}{m}-b n \ln \left (x \right ) \ln \left (\frac {d +e \,x^{m}}{d}\right )\) | \(180\) |
Input:
int((a+b*ln(c*(d+e*x^m)^n))/x,x,method=_RETURNVERBOSE)
Output:
b*ln(x)*ln((d+e*x^m)^n)+(1/2*I*b*Pi*csgn(I*(d+e*x^m)^n)*csgn(I*c*(d+e*x^m) ^n)^2-1/2*I*b*Pi*csgn(I*(d+e*x^m)^n)*csgn(I*c*(d+e*x^m)^n)*csgn(I*c)-1/2*I *b*Pi*csgn(I*c*(d+e*x^m)^n)^3+1/2*I*b*Pi*csgn(I*c*(d+e*x^m)^n)^2*csgn(I*c) +b*ln(c)+a)*ln(x)-b/m*n*dilog((d+e*x^m)/d)-b*n*ln(x)*ln((d+e*x^m)/d)
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\frac {b m n \log \left (e x^{m} + d\right ) \log \left (x\right ) - b m n \log \left (x\right ) \log \left (\frac {e x^{m} + d}{d}\right ) - b n {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (b m \log \left (c\right ) + a m\right )} \log \left (x\right )}{m} \] Input:
integrate((a+b*log(c*(d+e*x^m)^n))/x,x, algorithm="fricas")
Output:
(b*m*n*log(e*x^m + d)*log(x) - b*m*n*log(x)*log((e*x^m + d)/d) - b*n*dilog (-(e*x^m + d)/d + 1) + (b*m*log(c) + a*m)*log(x))/m
\[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\int \frac {a + b \log {\left (c \left (d + e x^{m}\right )^{n} \right )}}{x}\, dx \] Input:
integrate((a+b*ln(c*(d+e*x**m)**n))/x,x)
Output:
Integral((a + b*log(c*(d + e*x**m)**n))/x, x)
\[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^m)^n))/x,x, algorithm="maxima")
Output:
1/2*(2*d*m*n*integrate(log(x)/(e*x*x^m + d*x), x) - m*n*log(x)^2 + 2*log(( e*x^m + d)^n)*log(x) + 2*log(c)*log(x))*b + a*log(x)
\[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\int { \frac {b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^m)^n))/x,x, algorithm="giac")
Output:
integrate((b*log((e*x^m + d)^n*c) + a)/x, x)
Timed out. \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )}{x} \,d x \] Input:
int((a + b*log(c*(d + e*x^m)^n))/x,x)
Output:
int((a + b*log(c*(d + e*x^m)^n))/x, x)
\[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\left (x^{m} e +d \right )^{n} c \right )}{x^{m} e x +d x}d x \right ) b d m n +{\mathrm {log}\left (\left (x^{m} e +d \right )^{n} c \right )}^{2} b +2 \,\mathrm {log}\left (x \right ) a m n}{2 m n} \] Input:
int((a+b*log(c*(d+e*x^m)^n))/x,x)
Output:
(2*int(log((x**m*e + d)**n*c)/(x**m*e*x + d*x),x)*b*d*m*n + log((x**m*e + d)**n*c)**2*b + 2*log(x)*a*m*n)/(2*m*n)