\(\int \frac {\log (c (d+e x^{-n}))}{x (c e-(1-c d) x^n)} \, dx\) [380]

Optimal result

Integrand size = 33, antiderivative size = 26 \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,1-c \left (d+e x^{-n}\right )\right )}{c e n} \]

[Out]

polylog(2,1-c*(d+e/(x^n)))/c/e/n

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2525, 2459, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,1-c \left (e x^{-n}+d\right )\right )}{c e n} \]

[In]

Int[Log[c*(d + e/x^n)]/(x*(c*e - (1 - c*d)*x^n)),x]

[Out]

PolyLog[2, 1 - c*(d + e/x^n)]/(c*e*n)

Rule 2438

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]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[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]

Rule 2459

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol]
:> Int[(g + f*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x] && EqQ[m,
q] && IntegerQ[q]

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(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] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\log (c (d+e x))}{\left (c e+\frac {-1+c d}{x}\right ) x} \, dx,x,x^{-n}\right )}{n} \\ & = -\frac {\text {Subst}\left (\int \frac {\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^{-n}\right )}{n} \\ & = -\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,-1+c d+c e x^{-n}\right )}{c e n} \\ & = \frac {\text {Li}_2\left (1-c d-c e x^{-n}\right )}{c e n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,-x^{-n} \left (c e-x^n+c d x^n\right )\right )}{c e n} \]

[In]

Integrate[Log[c*(d + e/x^n)]/(x*(c*e - (1 - c*d)*x^n)),x]

[Out]

PolyLog[2, -((c*e - x^n + c*d*x^n)/x^n)]/(c*e*n)

Maple [A] (verified)

Time = 3.51 (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	\(\text {Expression too large to display}\)	\(1900\)

[In]

int(ln(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x,method=_RETURNVERBOSE)

[Out]

1/n/c/e*dilog(c*d+c*e/(x^n))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\frac {{\rm Li}_2\left (-\frac {c d x^{n} + c e}{x^{n}} + 1\right )}{c e n} \]

[In]

integrate(log(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x, algorithm="fricas")

[Out]

dilog(-(c*d*x^n + c*e)/x^n + 1)/(c*e*n)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\text {Timed out} \]

[In]

integrate(ln(c*(d+e/(x**n)))/x/(c*e-(-c*d+1)*x**n),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{n}}\right )}\right )}{{\left (c e + {\left (c d - 1\right )} x^{n}\right )} x} \,d x } \]

[In]

integrate(log(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x, algorithm="maxima")

[Out]

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)

Giac [F]

\[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{x^{n}}\right )}\right )}{{\left (c e + {\left (c d - 1\right )} x^{n}\right )} x} \,d x } \]

[In]

integrate(log(c*(d+e/(x^n)))/x/(c*e-(-c*d+1)*x^n),x, algorithm="giac")

[Out]

integrate(log(c*(d + e/x^n))/((c*e + (c*d - 1)*x^n)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx=\int \frac {\ln \left (c\,\left (d+\frac {e}{x^n}\right )\right )}{x\,\left (c\,e+x^n\,\left (c\,d-1\right )\right )} \,d x \]

[In]

int(log(c*(d + e/x^n))/(x*(c*e + x^n*(c*d - 1))),x)

[Out]

int(log(c*(d + e/x^n))/(x*(c*e + x^n*(c*d - 1))), x)