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

Optimal. Leaf size=50 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \text {Li}_2\left (\frac {e x^n}{d}+1\right )}{f n} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {12, 2454, 2394, 2315} \[ \frac {p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f n}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x^n)^p]/(f*x),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

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Mathematica [A]  time = 0.01, size = 46, normalized size = 0.92 \[ \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \text {Li}_2\left (\frac {e x^n+d}{d}\right )}{f n} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x^n)^p]/(f*x),x]

[Out]

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

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fricas [A]  time = 0.47, size = 63, normalized size = 1.26 \[ \frac {n p \log \left (e x^{n} + d\right ) \log \relax (x) - n p \log \relax (x) \log \left (\frac {e x^{n} + d}{d}\right ) + n \log \relax (c) \log \relax (x) - p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right )}{f n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((e*x^n + d)^p*c)/(f*x), x)

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maple [C]  time = 1.79, size = 201, normalized size = 4.02 \[ -\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right ) \ln \relax (x )}{2 f}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2 f}+\frac {i \pi \,\mathrm {csgn}\left (i \left (e \,x^{n}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{2} \ln \relax (x )}{2 f}-\frac {i \pi \mathrm {csgn}\left (i c \left (e \,x^{n}+d \right )^{p}\right )^{3} \ln \relax (x )}{2 f}-\frac {p \ln \relax (x ) \ln \left (\frac {e \,x^{n}+d}{d}\right )}{f}+\frac {\ln \relax (c ) \ln \relax (x )}{f}+\frac {\ln \relax (x ) \ln \left (\left (e \,x^{n}+d \right )^{p}\right )}{f}-\frac {p \dilog \left (\frac {e \,x^{n}+d}{d}\right )}{f n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*ln(x)*ln((e*x^n+d)^p)+1/2*I/f*ln(x)*Pi*csgn(I*(e*x^n+d)^p)*csgn(I*c*(e*x^n+d)^p)^2-1/2*I/f*ln(x)*Pi*csgn(I
*(e*x^n+d)^p)*csgn(I*c*(e*x^n+d)^p)*csgn(I*c)-1/2*I/f*ln(x)*Pi*csgn(I*c*(e*x^n+d)^p)^3+1/2*I/f*ln(x)*Pi*csgn(I
*c*(e*x^n+d)^p)^2*csgn(I*c)+1/f*ln(c)*ln(x)-1/f*p/n*dilog((e*x^n+d)/d)-1/f*p*ln(x)*ln((e*x^n+d)/d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^n)^p)/(f*x),x)

[Out]

int(log(c*(d + e*x^n)^p)/(f*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*(d + e*x**n)**p)/x, x)/f

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