3.159 \(\int \text{PolyLog}(n,d (F^{c (a+b x)})^p) \, dx\)

Optimal. Leaf size=31 \[ \frac{\text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]

[Out]

PolyLog[1 + n, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])

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Rubi [A]  time = 0.0177012, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2282, 6589} \[ \frac{\text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[n, d*(F^(c*(a + b*x)))^p],x]

[Out]

PolyLog[1 + n, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \text{Li}_n\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_n\left (d x^p\right )}{x} \, dx,x,F^{c (a+b x)}\right )}{b c \log (F)}\\ &=\frac{\text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.004743, size = 31, normalized size = 1. \[ \frac{\text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[n, d*(F^(c*(a + b*x)))^p],x]

[Out]

PolyLog[1 + n, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])

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Maple [A]  time = 0.061, size = 32, normalized size = 1. \begin{align*}{\frac{{\it polylog} \left ( 1+n,d \left ({F}^{c \left ( bx+a \right ) } \right ) ^{p} \right ) }{pbc\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(n,d*(F^(c*(b*x+a)))^p),x)

[Out]

polylog(1+n,d*(F^(c*(b*x+a)))^p)/b/c/p/ln(F)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d)\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,d*(F^(c*(b*x+a)))^p),x, algorithm="maxima")

[Out]

integrate(polylog(n, (F^((b*x + a)*c))^p*d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm polylog}\left (n,{\left (F^{b c x + a c}\right )}^{p} d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,d*(F^(c*(b*x+a)))^p),x, algorithm="fricas")

[Out]

integral(polylog(n, (F^(b*c*x + a*c))^p*d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Li}_{n}\left (d \left (F^{c \left (a + b x\right )}\right )^{p}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,d*(F**(c*(b*x+a)))**p),x)

[Out]

Integral(polylog(n, d*(F**(c*(a + b*x)))**p), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d)\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(n,d*(F^(c*(b*x+a)))^p),x, algorithm="giac")

[Out]

integrate(polylog(n, (F^((b*x + a)*c))^p*d), x)