\(\int \frac {\log (c (a-\frac {(d-a c d) x^{-m}}{c e}))}{x (d+e x^m)} \, dx\) [335]

Optimal result

Integrand size = 38, antiderivative size = 28 \[ \int \frac {\log \left (c \left (a-\frac {(d-a c d) x^{-m}}{c e}\right )\right )}{x \left (d+e x^m\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {(1-a c) \left (e+d x^{-m}\right )}{e}\right )}{d m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2525, 2459, 2440, 2438} \[ \int \frac {\log \left (c \left (a-\frac {(d-a c d) x^{-m}}{c e}\right )\right )}{x \left (d+e x^m\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]

[In]

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

[Out]

PolyLog[2, ((1 - a*c)*(e + d/x^m))/e]/(d*m)

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\log \left (c \left (a-\frac {(d-a c d) x^{-m}}{c e}\right )\right )}{x \left (d+e x^m\right )} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {(-1+a c) x^{-m} \left (d+e x^m\right )}{e}\right )}{d m} \]

[In]

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

[Out]

PolyLog[2, -(((-1 + a*c)*(d + e*x^m))/(e*x^m))]/(d*m)

Maple [A] (verified)

Time = 2.88 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {\operatorname {dilog}\left (\frac {\left (a c d -d \right ) x^{-m}}{e}+c a \right )}{m d}\)	\(30\)
default	\(\frac {\operatorname {dilog}\left (\frac {\left (a c d -d \right ) x^{-m}}{e}+c a \right )}{m d}\)	\(30\)
risch	\(\text {Expression too large to display}\)	\(1200\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {\log \left (c \left (a-\frac {(d-a c d) x^{-m}}{c e}\right )\right )}{x \left (d+e x^m\right )} \, dx=\frac {{\rm Li}_2\left (-\frac {a c e x^{m} + {\left (a c - 1\right )} d}{e x^{m}} + 1\right )}{d m} \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\log \left (c \left (a-\frac {(d-a c d) x^{-m}}{c e}\right )\right )}{x \left (d+e x^m\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(ln(c*(a+(a*c*d-d)/c/e/(x**m)))/x/(d+e*x**m),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

(a*c*m - m)*integrate(log(x)/(a*c*e*x*x^m + (a*c*d - d)*x), x) + (log(a*c*e*x^m + (a*c - 1)*d)*log(x) - log(e)
*log(x) - log(x)*log(x^m))/d + log(e)*log((e*x^m + d)/e)/(d*m) + (log(x^m)*log(e*x^m/d + 1) + dilog(-e*x^m/d))
/(d*m) - (log(a*c*e*x^m + (a*c - 1)*d)*log((a*c*e*x^m + a*c*d - d)/d + 1) + dilog(-(a*c*e*x^m + a*c*d - d)/d))
/(d*m)

Giac [F]

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

[In]

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

[Out]

integrate(log((a + (a*c*d - d)/(c*e*x^m))*c)/((e*x^m + d)*x), x)

Mupad [F(-1)]

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

[In]

int(log(c*(a - (d - a*c*d)/(c*e*x^m)))/(x*(d + e*x^m)),x)

[Out]

int(log(c*(a - (d - a*c*d)/(c*e*x^m)))/(x*(d + e*x^m)), x)