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

Optimal. Leaf size=39 \[ x \log (\pi )-\frac{\text{PolyLog}\left (2,-\frac{b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.0259307, antiderivative size = 39, normalized size of antiderivative = 1., 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 = {2279, 2392, 2391} \[ x \log (\pi )-\frac{\text{PolyLog}\left (2,-\frac{b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

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 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 2392

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]

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 (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx &=\frac{\operatorname{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{\operatorname{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]  time = 0.0086595, size = 39, normalized size = 1. \[ x \log (\pi )-\frac{\text{PolyLog}\left (2,-\frac{b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

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]  time = 0.02, size = 138, normalized size = 3.5 \begin{align*} -{\frac{1}{nde\ln \left ( F \right ) }\ln \left ( -{\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}}{\pi }} \right ) \ln \left ({\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}+\pi }{\pi }} \right ) }+{\frac{\ln \left ( b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}+\pi \right ) }{nde\ln \left ( F \right ) }\ln \left ( -{\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}}{\pi }} \right ) }-{\frac{1}{nde\ln \left ( F \right ) }{\it dilog} \left ({\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}+\pi }{\pi }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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

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]

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

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Fricas [B]  time = 2.14814, size = 250, normalized size = 6.41 \begin{align*} \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 )} \end{align*}

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*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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - 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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \log \left (\pi +{\left (F^{{\left (d x + c\right )} e}\right )}^{n} b\right )\,{d x} \end{align*}

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)