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

Optimal. Leaf size=41 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n}+\frac {\text {Li}_2\left (\frac {e x^n}{d}+1\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2454, 2394, 2315} \[ \frac {\text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)]/x,x]

[Out]

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 39, normalized size = 0.95 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )+\text {Li}_2\left (\frac {e x^n+d}{d}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^n)]/x,x]

[Out]

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

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fricas [A]  time = 0.45, size = 54, normalized size = 1.32 \[ \frac {n \log \left (c e x^{n} + c d\right ) \log \relax (x) - n \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) - {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )} c\right )}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((e*x^n + d)*c)/x, x)

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maple [A]  time = 0.14, size = 41, normalized size = 1.00 \[ \frac {\ln \left (-\frac {e \,x^{n}}{d}\right ) \ln \left (c e \,x^{n}+c d \right )}{n}+\frac {\dilog \left (-\frac {e \,x^{n}}{d}\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ d n \int \frac {\log \relax (x)}{e x x^{n} + d x}\,{d x} - \frac {1}{2} \, n \log \relax (x)^{2} + \log \left (e x^{n} + d\right ) \log \relax (x) + \log \relax (c) \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (c\,\left (d+e\,x^n\right )\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^n))/x,x)

[Out]

int(log(c*(d + e*x^n))/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c d + c e x^{n} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*d + c*e*x**n)/x, x)

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