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

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 = 0.03, 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 = {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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 1.00 \[ x \log (\pi )-\frac {\text {Li}_2\left (-\frac {b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]

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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fricas [B]  time = 0.73, size = 106, normalized size = 2.72 \[ \frac {{\left (d e n x + c e n\right )} \log \left (\pi + F^{d e n x + c e n} b\right ) \log \relax (F) - {\left (d e n x + c e n\right )} \log \relax (F) \log \left (\frac {\pi + F^{d e n x + c e n} b}{\pi }\right ) - {\rm Li}_2\left (-\frac {\pi + F^{d e n x + c e n} b}{\pi } + 1\right )}{d e n \log \relax (F)} \]

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

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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maple [B]  time = 0.15, size = 138, normalized size = 3.54 \[ -\frac {\ln \left (\frac {b \left (F^{\left (d x +c \right ) e}\right )^{n}+\pi }{\pi }\right ) \ln \left (-\frac {b \left (F^{\left (d x +c \right ) e}\right )^{n}}{\pi }\right )}{d e n \ln \relax (F )}+\frac {\ln \left (-\frac {b \left (F^{\left (d x +c \right ) e}\right )^{n}}{\pi }\right ) \ln \left (b \left (F^{\left (d x +c \right ) e}\right )^{n}+\pi \right )}{d e n \ln \relax (F )}-\frac {\dilog \left (\frac {b \left (F^{\left (d x +c \right ) e}\right )^{n}+\pi }{\pi }\right )}{d e n \ln \relax (F )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, d e n x^{2} \log \relax (F) + \pi d e n \int \frac {x}{\pi + {\left (F^{d e x}\right )}^{n} {\left (F^{c e}\right )}^{n} b}\,{d x} \log \relax (F) + x \log \left (\pi + {\left (F^{d e x}\right )}^{n} {\left (F^{c e}\right )}^{n} b\right ) \]

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

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

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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