∫ log (x−m(−d+acd+acexm) e ) _________________________________

Optimal. Leaf size=28 \[ \frac {\text {Li}_2\left (\frac {(1-a c) \left (e+d x^{-m}\right )}{e}\right )}{d m} \]

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Rubi [A]
time = 0.12, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2530, 2525, 2459, 2440, 2438} \begin {gather*} \frac {\text {PolyLog}\left (2,\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \end {gather*}

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

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,

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

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,

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])

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*(u_)^(r_.)*((h_.)*(x_))^(m_.), x_Symbol] :> Int[(h*x)^m*Expand

ToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, h, m, p, q, r}, x] && BinomialQ[{u,

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

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.11 \begin {gather*} \frac {\text {Li}_2\left (-\frac {(-1+a c) x^{-m} \left (d+e x^m\right )}{e}\right )}{d m} \end {gather*}

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

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method	result	size

derivativedivides	\(\frac {\dilog \left (c a +\frac {d \left (c a -1\right ) x^{-m}}{e}\right )}{m d}\)	\(28\)
default	\(\frac {\dilog \left (c a +\frac {d \left (c a -1\right ) x^{-m}}{e}\right )}{m d}\)	\(28\)
risch	\(\text {Expression too large to display}\)	\(1200\)

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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

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

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

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Fricas [A]
time = 0.36, size = 35, normalized size = 1.25 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {{\left (a c x^{m} e + {\left (a c - 1\right )} d\right )} e^{\left (-1\right )}}{x^{m}} + 1\right )}{d m} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (\frac {a\,c\,d-d+a\,c\,e\,x^m}{e\,x^m}\right )}{x\,\left (d+e\,x^m\right )} \,d x \end {gather*}

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

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

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3.4.36 \(\int \frac {\log (\frac {x^{-m} (-d+a c d+a c e x^m)}{e})}{x (d+e x^m)} \, dx\) [336]