\(\int \frac {x^{-1+n} \log (c (d+e x^n))}{-1+c d+c e x^n} \, dx\) [379]

Optimal result

Integrand size = 29, antiderivative size = 25 \[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=-\frac {\operatorname {PolyLog}\left (2,1-c \left (d+e x^n\right )\right )}{c e n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2525, 2440, 2438} \[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=-\frac {\operatorname {PolyLog}\left (2,1-c \left (e x^n+d\right )\right )}{c e n} \]

[In]

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

[Out]

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

Rule 2438

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]

Rule 2440

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=-\frac {\operatorname {PolyLog}\left (2,1-c d-c e x^n\right )}{c e n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.69 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {\operatorname {dilog}\left (c e \,x^{n}+c d \right )}{n c e}\)	\(23\)
risch	\(\frac {\ln \left (1-c \left (d +e \,x^{n}\right )\right ) \ln \left (d +e \,x^{n}\right )}{n e c}-\frac {\ln \left (1-c \left (d +e \,x^{n}\right )\right ) \ln \left (c \left (d +e \,x^{n}\right )\right )}{n e c}-\frac {\operatorname {dilog}\left (c \left (d +e \,x^{n}\right )\right )}{n e c}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (-1+c d +c e \,x^{n}\right )}{n c e}\)	\(215\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=-\frac {{\rm Li}_2\left (-c e x^{n} - c d + 1\right )}{c e n} \]

[In]

integrate(x^(-1+n)*log(c*(d+e*x^n))/(-1+c*d+c*e*x^n),x, algorithm="fricas")

[Out]

-dilog(-c*e*x^n - c*d + 1)/(c*e*n)

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (24) = 48\).

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.36 \[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=\frac {\log \left (c e x^{n} + c d - 1\right ) \log \left ({\left (e x^{n} + d\right )} c\right )}{c e n} - \frac {\log \left (c e x^{n} + c d - 1\right ) \log \left (e x^{n} + d\right )}{c e n} + \frac {\log \left (-c e x^{n} - c d + 1\right ) \log \left (e x^{n} + d\right ) + {\rm Li}_2\left (c e x^{n} + c d\right )}{c e n} \]

[In]

integrate(x^(-1+n)*log(c*(d+e*x^n))/(-1+c*d+c*e*x^n),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=\int { \frac {x^{n - 1} \log \left ({\left (e x^{n} + d\right )} c\right )}{c e x^{n} + c d - 1} \,d x } \]

[In]

integrate(x^(-1+n)*log(c*(d+e*x^n))/(-1+c*d+c*e*x^n),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.88 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx=-\frac {{\mathrm {Li}}_{\mathrm {2}}\left (c\,\left (d+e\,x^n\right )\right )}{c\,e\,n} \]

[In]

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

[Out]

-dilog(c*(d + e*x^n))/(c*e*n)