\(\int \log (b (F^{e (c+d x)})^n+\pi ) \, dx\) [128]

Optimal result

Integrand size = 16, antiderivative size = 39 \[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=x \log (\pi )-\frac {\operatorname {PolyLog}\left (2,-\frac {b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

[Out]

x*ln(Pi)-polylog(2,-b*(F^(e*(d*x+c)))^n/Pi)/d/e/n/ln(F)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2317, 2439, 2438} \[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=x \log (\pi )-\frac {\operatorname {PolyLog}\left (2,-\frac {b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

[In]

Int[Log[b*(F^(e*(c + d*x)))^n + Pi],x]

[Out]

x*Log[Pi] - PolyLog[2, -((b*(F^(e*(c + d*x)))^n)/Pi)]/(d*e*n*Log[F])

Rule 2317

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 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 2439

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\log (\pi +b x)}{x} \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)} \\ & = x \log (\pi )+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{\pi }\right )}{x} \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)} \\ & = x \log (\pi )-\frac {\text {Li}_2\left (-\frac {b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=x \log (\pi )-\frac {\operatorname {PolyLog}\left (2,-\frac {b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

[In]

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

[Out]

x*Log[Pi] - PolyLog[2, -((b*(F^(e*(c + d*x)))^n)/Pi)]/(d*e*n*Log[F])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(39)=78\).

Time = 1.40 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.46

method	result	size
derivativedivides	\(\frac {\left (\ln \left (b \left (F^{e \left (d x +c \right )}\right )^{n}+\pi \right )-\ln \left (\frac {b \left (F^{e \left (d x +c \right )}\right )^{n}+\pi }{\pi }\right )\right ) \ln \left (-\frac {b \left (F^{e \left (d x +c \right )}\right )^{n}}{\pi }\right )-\operatorname {dilog}\left (\frac {b \left (F^{e \left (d x +c \right )}\right )^{n}+\pi }{\pi }\right )}{d e \ln \left (F \right ) n}\)	\(96\)
default	\(\frac {\left (\ln \left (b \left (F^{e \left (d x +c \right )}\right )^{n}+\pi \right )-\ln \left (\frac {b \left (F^{e \left (d x +c \right )}\right )^{n}+\pi }{\pi }\right )\right ) \ln \left (-\frac {b \left (F^{e \left (d x +c \right )}\right )^{n}}{\pi }\right )-\operatorname {dilog}\left (\frac {b \left (F^{e \left (d x +c \right )}\right )^{n}+\pi }{\pi }\right )}{d e \ln \left (F \right ) n}\)	\(96\)
risch	\(x \ln \left (b \left (F^{e \left (d x +c \right )}\right )^{n}+\pi \right )-\frac {\operatorname {dilog}\left (\frac {b \,F^{x n e d} F^{-x n e d} \left (F^{e \left (d x +c \right )}\right )^{n}+\pi }{\pi }\right )}{\ln \left (F \right ) d e n}-\frac {\ln \left (F^{e \left (d x +c \right )}\right ) \ln \left (\frac {b \,F^{x n e d} F^{-x n e d} \left (F^{e \left (d x +c \right )}\right )^{n}+\pi }{\pi }\right )}{\ln \left (F \right ) d e}-\ln \left (b \,F^{x n e d} F^{-x n e d} \left (F^{e \left (d x +c \right )}\right )^{n}+\pi \right ) x +\frac {\ln \left (b \,F^{x n e d} F^{-x n e d} \left (F^{e \left (d x +c \right )}\right )^{n}+\pi \right ) \ln \left (F^{e \left (d x +c \right )}\right )}{\ln \left (F \right ) d e}\)	\(213\)

[In]

int(ln(b*(F^(e*(d*x+c)))^n+Pi),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (38) = 76\).

Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.72 \[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=\frac {{\left (d e n x + c e n\right )} \log \left (\pi + F^{d e n x + c e n} b\right ) \log \left (F\right ) - {\left (d e n x + c e n\right )} \log \left (F\right ) \log \left (\frac {\pi + F^{d e n x + c e n} b}{\pi }\right ) - {\rm Li}_2\left (-\frac {\pi + F^{d e n x + c e n} b}{\pi } + 1\right )}{d e n \log \left (F\right )} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=\int \log {\left (b \left (F^{e \left (c + d x\right )}\right )^{n} + \pi \right )}\, dx \]

[In]

integrate(ln(b*(F**(e*(d*x+c)))**n+pi),x)

[Out]

Integral(log(b*(F**(e*(c + d*x)))**n + pi), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (38) = 76\).

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.10 \[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=x \log \left (\pi + F^{{\left (d x + c\right )} e n} b\right ) - \frac {d e n x \log \left (\frac {F^{d e n x} F^{c e n} b}{\pi } + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{d e n x} F^{c e n} b}{\pi }\right )}{d e n \log \left (F\right )} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=\int { \log \left (\pi + {\left (F^{{\left (d x + c\right )} e}\right )}^{n} b\right ) \,d x } \]

[In]

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

[Out]

integrate(log(pi + (F^((d*x + c)*e))^n*b), x)

Mupad [F(-1)]

Timed out. \[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx=\int \ln \left (\Pi +b\,{\left (F^{e\,\left (c+d\,x\right )}\right )}^n\right ) \,d x \]

[In]

int(log(Pi + b*(F^(e*(c + d*x)))^n),x)

[Out]

int(log(Pi + b*(F^(e*(c + d*x)))^n), x)