3.4.63 \(\int \frac {(f+g x^{-n}) \log (c (d+e x^n)^p)}{x} \, dx\) [363]

Optimal. Leaf size=97 \[ \frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^n\right )}{d n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2525, 14, 2463, 2442, 36, 29, 31, 2441, 2352} \begin {gather*} \frac {f p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {e g p \log \left (d+e x^n\right )}{d n}+\frac {e g p \log (x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2352

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

Rule 2441

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 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2525

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])

Rubi steps

\begin {align*} \int \frac {\left (f+g x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (f+\frac {g}{x}\right ) \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {g \log \left (c (d+e x)^p\right )}{x^2}+\frac {f \log \left (c (d+e x)^p\right )}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {g \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {(e f p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^n\right )}{n}\\ &=-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}+\frac {(e g p) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{d n}-\frac {\left (e^2 g p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{d n}\\ &=\frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^n\right )}{d n}-\frac {g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 100, normalized size = 1.03 \begin {gather*} -\frac {e g p \log \left (d n \left (d+e x^n\right )\right )+d g x^{-n} \log \left (c \left (d+e x^n\right )^p\right )-n \log (x) \left (e g p+d f \log \left (c \left (d+e x^n\right )^p\right )-d f p \log \left (1+\frac {e x^n}{d}\right )\right )+d f p \text {Li}_2\left (-\frac {e x^n}{d}\right )}{d n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 423, normalized size = 4.36

method	result	size
risch	\(\frac {\left (f \ln \left (x \right ) n \,x^{n}-g \right ) x^{-n} \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} f \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} g \,x^{-n}}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) f \ln \left (x^{n}\right )}{2 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) g \,x^{-n}}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f \ln \left (x^{n}\right )}{2 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} g \,x^{-n}}{2 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) g \,x^{-n}}{2 n}+\frac {\ln \left (c \right ) f \ln \left (x^{n}\right )}{n}-\frac {\ln \left (c \right ) g \,x^{-n}}{n}-\frac {p f \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n}-p f \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )-\frac {e g p \ln \left (d +e \,x^{n}\right )}{d n}+\frac {p e g \ln \left (x^{n}\right )}{n d}\)	\(423\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 119, normalized size = 1.23 \begin {gather*} -\frac {d f n p x^{n} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) + d f p x^{n} {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) + d g \log \left (c\right ) - {\left (g n p e + d f n \log \left (c\right )\right )} x^{n} \log \left (x\right ) + {\left (d g p - {\left (d f n p \log \left (x\right ) - g p e\right )} x^{n}\right )} \log \left (x^{n} e + d\right )}{d n x^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{- n} \left (f x^{n} + g\right ) \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+\frac {g}{x^n}\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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