Integrand size = 23, antiderivative size = 39 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d} \] Output:
(a+b*ln(c*x^n))*ln(1+d*x/e)/d+b*n*polylog(2,-d*x/e)/d
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )+b n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d} \] Input:
Integrate[(a + b*Log[c*x^n])/((d + e/x)*x),x]
Output:
((a + b*Log[c*x^n])*Log[1 + (d*x)/e] + b*n*PolyLog[2, -((d*x)/e)])/d
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2005, 2754, 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.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x \left (d+\frac {e}{x}\right )} \, dx\) |
\(\Big \downarrow \) 2005 |
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{d x+e}dx\) |
\(\Big \downarrow \) 2754 |
\(\displaystyle \frac {\log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {b n \int \frac {\log \left (\frac {d x}{e}+1\right )}{x}dx}{d}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d}\) |
Input:
Int[(a + b*Log[c*x^n])/((d + e/x)*x),x]
Output:
((a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d + (b*n*PolyLog[2, -((d*x)/e)])/d
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[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.87
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) \ln \left (d x +e \right )}{d}-\frac {b n \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{d}-\frac {b n \operatorname {dilog}\left (-\frac {d x}{e}\right )}{d}+\frac {\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \ln \left (d x +e \right )}{d}\) | \(151\) |
Input:
int((a+b*ln(c*x^n))/(d+e/x)/x,x,method=_RETURNVERBOSE)
Output:
b*ln(x^n)*ln(d*x+e)/d-b/d*n*ln(d*x+e)*ln(-d*x/e)-b/d*n*dilog(-d*x/e)+(1/2* I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*cs gn(I*c)-1/2*I*Pi*b*csgn(I*c*x^n)^3+1/2*I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+b* ln(c)+a)*ln(d*x+e)/d
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(d+e/x)/x,x, algorithm="fricas")
Output:
integral((b*log(c*x^n) + a)/(d*x + e), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{d x + e}\, dx \] Input:
integrate((a+b*ln(c*x**n))/(d+e/x)/x,x)
Output:
Integral((a + b*log(c*x**n))/(d*x + e), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(d+e/x)/x,x, algorithm="maxima")
Output:
b*integrate((log(c) + log(x^n))/(d*x + e), x) + a*log(d*x + e)/d
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(d+e/x)/x,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)/((d + e/x)*x), x)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+\frac {e}{x}\right )} \,d x \] Input:
int((a + b*log(c*x^n))/(x*(d + e/x)),x)
Output:
int((a + b*log(c*x^n))/(x*(d + e/x)), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx=\frac {-2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d \,x^{2}+e x}d x \right ) b e n +2 \,\mathrm {log}\left (d x +e \right ) a n +\mathrm {log}\left (x^{n} c \right )^{2} b}{2 d n} \] Input:
int((a+b*log(c*x^n))/(d+e/x)/x,x)
Output:
( - 2*int(log(x**n*c)/(d*x**2 + e*x),x)*b*e*n + 2*log(d*x + e)*a*n + log(x **n*c)**2*b)/(2*d*n)