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

Optimal. Leaf size=28 \[ \frac{\text{PolyLog}\left (2,\frac{(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]

[Out]

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

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Rubi [A]  time = 0.134979, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2475, 2412, 2393, 2391} \[ \frac{\text{PolyLog}\left (2,\frac{(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]

Antiderivative was successfully verified.

[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 2475

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])

Rule 2412

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 2393

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 2391

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]

Rubi steps

\begin{align*} \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{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{\operatorname{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{\operatorname{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]  time = 0.0228317, size = 31, normalized size = 1.11 \[ \frac{\text{PolyLog}\left (2,-\frac{(a c-1) x^{-m} \left (d+e x^m\right )}{e}\right )}{d m} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.065, size = 28, normalized size = 1. \begin{align*}{\frac{1}{md}{\it dilog} \left ( ac+{\frac{d \left ( ac-1 \right ) }{e{x}^{m}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (a c m - m\right )} \int \frac{\log \left (x\right )}{a c e x x^{m} +{\left (a c d - d\right )} x}\,{d x} + \frac{\log \left (a c e x^{m} +{\left (a c - 1\right )} d\right ) \log \left (x\right ) - \log \left (e\right ) \log \left (x\right ) - \log \left (x\right ) \log \left (x^{m}\right )}{d} + \frac{\log \left (e\right ) \log \left (\frac{e x^{m} + d}{e}\right )}{d m} + \frac{\log \left (x^{m}\right ) \log \left (\frac{e x^{m}}{d} + 1\right ) +{\rm Li}_2\left (-\frac{e x^{m}}{d}\right )}{d m} - \frac{\log \left (a c e x^{m} +{\left (a c - 1\right )} d\right ) \log \left (\frac{a c e x^{m} + a c d - d}{d} + 1\right ) +{\rm Li}_2\left (-\frac{a c e x^{m} + a c d - d}{d}\right )}{d m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.74844, size = 72, normalized size = 2.57 \begin{align*} \frac{{\rm Li}_2\left (-\frac{a c e x^{m} +{\left (a c - 1\right )} d}{e x^{m}} + 1\right )}{d m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)