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

Optimal. Leaf size=31 \[ -\frac {\text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]

[Out]

-polylog(2,-e*(f^(c*(b*x+a)))^n)/b/c/n/ln(f)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 = 0.00, size = 31, normalized size = 1.00 \[ -\frac {\text {Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]

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]))

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 31, normalized size = 1.00 \[ -\frac {{\rm Li}_2\left (-e f^{b c n x + a c n}\right )}{b c n \log \relax (f)} \]

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))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (e {\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right )\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 32, normalized size = 1.03 \[ -\frac {\dilog \left (e \left (f^{\left (b x +a \right ) c}\right )^{n}+1\right )}{b c n \ln \relax (f )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.90, size = 76, normalized size = 2.45 \[ x \log \left (e f^{{\left (b x + a\right )} c n} + 1\right ) - \frac {b c n x \log \left (e f^{b c n x} f^{a c n} + 1\right ) \log \relax (f) + {\rm Li}_2\left (-e f^{b c n x} f^{a c n}\right )}{b c n \log \relax (f)} \]

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]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \ln \left (e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n+1\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(e*(f^(c*(a + b*x)))^n + 1),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - b c e n e^{a c n \log {\relax (f )}} \log {\relax (f )} \int \frac {x e^{b c n x \log {\relax (f )}}}{e e^{a c n \log {\relax (f )}} e^{b c n x \log {\relax (f )}} + 1}\, dx + x \log {\left (e \left (f^{c \left (a + b x\right )}\right )^{n} + 1 \right )} \]

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)

________________________________________________________________________________________