Integrand size = 33, antiderivative size = 26 \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,1-c \left (d+e x^{-n}\right )\right )}{c e n} \] Output:
polylog(2,1-c*(d+e/(x^n)))/c/e/n
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,-x^{-n} \left (c e-x^n+c d x^n\right )\right )}{c e n} \] Input:
Integrate[Log[c*(d + e/x^n)]/(x*(c*e - (1 - c*d)*x^n)),x]
Output:
PolyLog[2, -((c*e - x^n + c*d*x^n)/x^n)]/(c*e*n)
Time = 0.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2925, 2005, 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.
\(\displaystyle \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx\) |
\(\Big \downarrow \) 2925 |
\(\displaystyle -\frac {\int \frac {x^n \log \left (c \left (e x^{-n}+d\right )\right )}{c e-(1-c d) x^n}dx^{-n}}{n}\) |
\(\Big \downarrow \) 2005 |
\(\displaystyle -\frac {\int \frac {\log \left (c \left (e x^{-n}+d\right )\right )}{c e x^{-n}+c d-1}dx^{-n}}{n}\) |
\(\Big \downarrow \) 2840 |
\(\displaystyle -\frac {\int x^n \log \left (c e x^{-n}+c d\right )d\left (c e x^{-n}+c d-1\right )}{c e n}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\operatorname {PolyLog}\left (2,-c e x^{-n}-c d+1\right )}{c e n}\) |
Input:
Int[Log[c*(d + e/x^n)]/(x*(c*e - (1 - c*d)*x^n)),x]
Output:
PolyLog[2, 1 - c*d - (c*e)/x^n]/(c*e*n)
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_))^(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])
Time = 8.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\operatorname {dilog}\left (c d +c e \,x^{-n}\right )}{n c e}\) | \(24\) |
default | \(\frac {\operatorname {dilog}\left (c d +c e \,x^{-n}\right )}{n c e}\) | \(24\) |
risch | \(\text {Expression too large to display}\) | \(1900\) |
Input:
int(ln(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x,method=_RETURNVERBOSE)
Output:
1/n*dilog(c*d+c*e/(x^n))/c/e
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {{\rm Li}_2\left (-\frac {c d x^{n} + c e}{x^{n}} + 1\right )}{c e n} \] Input:
integrate(log(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x, algorithm="fricas")
Output:
dilog(-(c*d*x^n + c*e)/x^n + 1)/(c*e*n)
Timed out. \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\text {Timed out} \] Input:
integrate(ln(c*(d+e/(x**n)))/x/(c*e-(-c*d+1)*x**n),x)
Output:
Timed out
\[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{n}}\right )}\right )}{{\left (c e + {\left (c d - 1\right )} x^{n}\right )} x} \,d x } \] Input:
integrate(log(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x, algorithm="maxima")
Output:
n*integrate(log(x)/(c*d*x*x^n + c*e*x), x) + (log(d*x^n + e)*log(x) + log( c)*log(x) - log(x)*log(x^n))/(c*e) - log(c)*log((c*e + (c*d - 1)*x^n)/(c*d - 1))/(c*e*n) - (log(d*x^n + e)*log((c*d*e + (c*d^2 - d)*x^n - e)/e + 1) + dilog(-(c*d*e + (c*d^2 - d)*x^n - e)/e))/(c*e*n) + (log(x^n)*log((c*d - 1)*x^n/(c*e) + 1) + dilog(-(c*d - 1)*x^n/(c*e)))/(c*e*n)
\[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{n}}\right )}\right )}{{\left (c e + {\left (c d - 1\right )} x^{n}\right )} x} \,d x } \] Input:
integrate(log(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x, algorithm="giac")
Output:
integrate(log(c*(d + e/x^n))/((c*e + (c*d - 1)*x^n)*x), x)
Timed out. \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\int \frac {\ln \left (c\,\left (d+\frac {e}{x^n}\right )\right )}{x\,\left (c\,e+x^n\,\left (c\,d-1\right )\right )} \,d x \] Input:
int(log(c*(d + e/x^n))/(x*(c*e + x^n*(c*d - 1))),x)
Output:
int(log(c*(d + e/x^n))/(x*(c*e + x^n*(c*d - 1))), x)
\[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {-2 \left (\int \frac {\mathrm {log}\left (\frac {x^{n} c d +c e}{x^{n}}\right )}{x^{2 n} c^{2} d^{3} x -2 x^{2 n} c \,d^{2} x +x^{2 n} d x +2 x^{n} c^{2} d^{2} e x -3 x^{n} c d e x +x^{n} e x +c^{2} d \,e^{2} x -c \,e^{2} x}d x \right ) c d \,e^{2} n +2 \left (\int \frac {\mathrm {log}\left (\frac {x^{n} c d +c e}{x^{n}}\right )}{x^{2 n} c^{2} d^{3} x -2 x^{2 n} c \,d^{2} x +x^{2 n} d x +2 x^{n} c^{2} d^{2} e x -3 x^{n} c d e x +x^{n} e x +c^{2} d \,e^{2} x -c \,e^{2} x}d x \right ) e^{2} n -\mathrm {log}\left (\frac {x^{n} c d +c e}{x^{n}}\right )^{2} d}{2 e n \left (c d -1\right )} \] Input:
int(log(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x)
Output:
( - 2*int(log((x**n*c*d + c*e)/x**n)/(x**(2*n)*c**2*d**3*x - 2*x**(2*n)*c* d**2*x + x**(2*n)*d*x + 2*x**n*c**2*d**2*e*x - 3*x**n*c*d*e*x + x**n*e*x + c**2*d*e**2*x - c*e**2*x),x)*c*d*e**2*n + 2*int(log((x**n*c*d + c*e)/x**n )/(x**(2*n)*c**2*d**3*x - 2*x**(2*n)*c*d**2*x + x**(2*n)*d*x + 2*x**n*c**2 *d**2*e*x - 3*x**n*c*d*e*x + x**n*e*x + c**2*d*e**2*x - c*e**2*x),x)*e**2* n - log((x**n*c*d + c*e)/x**n)**2*d)/(2*e*n*(c*d - 1))