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

Optimal. Leaf size=39 \[ x \log (\pi )-\frac {\text {Li}_2\left (-\frac {b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2317, 2439, 2438} \begin {gather*} x \log (\pi )-\frac {\text {PolyLog}\left (2,-\frac {b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.00 \begin {gather*} 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 {gather*}

Antiderivative was successfully verified.

[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])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(39)=78\).
time = 0.08, size = 96, normalized size = 2.46

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (38) = 76\).
time = 0.29, size = 87, normalized size = 2.23 \begin {gather*} x \log \left (\pi + F^{{\left (d x + c\right )} n e} b\right ) - \frac {{\left (d n x e \log \left (\frac {F^{d n x e} F^{c n e} b}{\pi } + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{d n x e} F^{c n e} b}{\pi }\right )\right )} e^{\left (-1\right )}}{d n \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (38) = 76\).
time = 0.37, size = 108, normalized size = 2.77 \begin {gather*} \frac {{\left ({\left (d n x + c n\right )} e \log \left (\pi + F^{{\left (d n x + c n\right )} e} b\right ) \log \left (F\right ) - {\left (d n x + c n\right )} e \log \left (F\right ) \log \left (\frac {\pi + F^{{\left (d n x + c n\right )} e} b}{\pi }\right ) - {\rm Li}_2\left (-\frac {\pi + F^{{\left (d n x + c n\right )} e} b}{\pi } + 1\right )\right )} e^{\left (-1\right )}}{d n \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - b d e n e^{c e n \log {\left (F \right )}} \log {\left (F \right )} \int \frac {x e^{d e n x \log {\left (F \right )}}}{b e^{c e n \log {\left (F \right )}} e^{d e n x \log {\left (F \right )}} + \pi }\, dx + x \log {\left (b \left (F^{e \left (c + d x\right )}\right )^{n} + \pi \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \ln \left (\Pi +b\,{\left (F^{e\,\left (c+d\,x\right )}\right )}^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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