### 3.121 $$\int \log (1+e (f^{c (a+b x)})^n) \, dx$$

Optimal. Leaf size=31 $-\frac{\text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0145746, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {2279, 2391} $-\frac{\text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[1 + e*(f^(c*(a + b*x)))^n],x]

[Out]

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (1+e x)}{x} \, dx,x,\left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}\\ &=-\frac{\text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}\\ \end{align*}

Mathematica [A]  time = 0.0012386, size = 31, normalized size = 1. $-\frac{\text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[1 + e*(f^(c*(a + b*x)))^n],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 32, normalized size = 1. \begin{align*} -{\frac{{\it dilog} \left ( 1+e \left ({f}^{c \left ( bx+a \right ) } \right ) ^{n} \right ) }{ncb\ln \left ( f \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(1+e*(f^(c*(b*x+a)))^n),x)

[Out]

-1/c/b/ln(f)/n*dilog(1+e*(f^(c*(b*x+a)))^n)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1+e*(f^(c*(b*x+a)))^n),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.19991, size = 63, normalized size = 2.03 \begin{align*} -\frac{{\rm Li}_2\left (-e f^{b c n x + a c n}\right )}{b c n \log \left (f\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1+e*(f^(c*(b*x+a)))^n),x, algorithm="fricas")

[Out]

-dilog(-e*f^(b*c*n*x + a*c*n))/(b*c*n*log(f))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - b c e n e^{a c n \log{\left (f \right )}} \log{\left (f \right )} \int \frac{x e^{b c n x \log{\left (f \right )}}}{e e^{a c n \log{\left (f \right )}} e^{b c n x \log{\left (f \right )}} + 1}\, dx + x \log{\left (e \left (f^{c \left (a + b x\right )}\right )^{n} + 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(1+e*(f**(c*(b*x+a)))**n),x)

[Out]

-b*c*e*n*exp(a*c*n*log(f))*log(f)*Integral(x*exp(b*c*n*x*log(f))/(e*exp(a*c*n*log(f))*exp(b*c*n*x*log(f)) + 1)
, x) + x*log(e*(f**(c*(a + b*x)))**n + 1)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (e{\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1+e*(f^(c*(b*x+a)))^n),x, algorithm="giac")

[Out]

integrate(log(e*(f^((b*x + a)*c))^n + 1), x)