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

Optimal. Leaf size=26 \[ \frac{\text{PolyLog}\left (2,1-c \left (d+e x^{-n}\right )\right )}{c e n} \]

[Out]

PolyLog[2, 1 - c*(d + e/x^n)]/(c*e*n)

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Rubi [A]  time = 0.155378, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {2475, 2412, 2393, 2391} \[ \frac{\text{PolyLog}\left (2,1-c \left (d+e x^{-n}\right )\right )}{c e n} \]

Antiderivative was successfully verified.

[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 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 (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx &=-\frac{\operatorname{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{\operatorname{Subst}\left (\int \frac{\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^{-n}\right )}{n}\\ &=-\frac{\operatorname{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]  time = 0.068676, size = 34, normalized size = 1.31 \[ \frac{\text{PolyLog}\left (2,-x^{-n} \left (c d x^n+c e-x^n\right )\right )}{c e n} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.088, size = 24, normalized size = 0.9 \begin{align*}{\frac{1}{cen}{\it dilog} \left ( cd+{\frac{ce}{{x}^{n}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} n \int \frac{\log \left (x\right )}{c d x x^{n} + c e x}\,{d x} + \frac{\log \left (d x^{n} + e\right ) \log \left (x\right ) + \log \left (c\right ) \log \left (x\right ) - \log \left (x\right ) \log \left (x^{n}\right )}{c e} - \frac{\log \left (c\right ) \log \left (\frac{c e +{\left (c d - 1\right )} x^{n}}{c d - 1}\right )}{c e n} - \frac{\log \left (d x^{n} + e\right ) \log \left (\frac{c d e +{\left (c d^{2} - d\right )} x^{n} - e}{e} + 1\right ) +{\rm Li}_2\left (-\frac{c d e +{\left (c d^{2} - d\right )} x^{n} - e}{e}\right )}{c e n} + \frac{\log \left (x^{n}\right ) \log \left (\frac{{\left (c d - 1\right )} x^{n}}{c e} + 1\right ) +{\rm Li}_2\left (-\frac{{\left (c d - 1\right )} x^{n}}{c e}\right )}{c e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.75127, size = 55, normalized size = 2.12 \begin{align*} \frac{{\rm Li}_2\left (-\frac{c d x^{n} + c e}{x^{n}} + 1\right )}{c e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)