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

Optimal. Leaf size=75 \[ -\frac{\text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+x \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-x \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.127399, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2280, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+x \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-x \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[d + e*(f^(c*(a + b*x)))^n],x]

[Out]

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

Rule 2280

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[x*Log[a + b*(F^(e*(c + d*x)
))^n], x] - Dist[b*d*e*n*Log[F], Int[(x*(F^(e*(c + d*x)))^n)/(a + b*(F^(e*(c + d*x)))^n), x], x] /; FreeQ[{F,
a, b, c, d, e, n}, x] &&  !GtQ[a, 0]

Rule 2190

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

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

Mathematica [A]  time = 0.0030972, size = 75, normalized size = 1. \[ -\frac{\text{PolyLog}\left (2,-\frac{e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+x \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-x \log \left (\frac{e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 82, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( d+e \left ({f}^{c \left ( bx+a \right ) } \right ) ^{n} \right ) }{ncb\ln \left ( f \right ) }\ln \left ( -{\frac{e \left ({f}^{c \left ( bx+a \right ) } \right ) ^{n}}{d}} \right ) }+{\frac{1}{ncb\ln \left ( f \right ) }{\it dilog} \left ( -{\frac{e \left ({f}^{c \left ( bx+a \right ) } \right ) ^{n}}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 d n \int \frac{x}{e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + d}\,{d x} \log \left (f\right ) + x \log \left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.1413, size = 243, normalized size = 3.24 \begin{align*} \frac{{\left (b c n x + a c n\right )} \log \left (e f^{b c n x + a c n} + d\right ) \log \left (f\right ) -{\left (b c n x + a c n\right )} \log \left (f\right ) \log \left (\frac{e f^{b c n x + a c n} + d}{d}\right ) -{\rm Li}_2\left (-\frac{e f^{b c n x + a c n} + d}{d} + 1\right )}{b c n \log \left (f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}}}{d + e e^{a c n \log{\left (f \right )}} e^{b c n x \log{\left (f \right )}}}\, dx + x \log{\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(d+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))/(d + e*exp(a*c*n*log(f))*exp(b*c*n*x*log(f)))
, x) + x*log(d + e*(f**(c*(a + b*x)))**n)

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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} + d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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