3.2.59 \(\int \text {PolyLog}(n,d (F^{c (a+b x)})^p) \, dx\) [159]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 = {2320, 6724} \begin {gather*} \frac {\text {Li}_{n+1}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \end {gather*}

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 2320

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 6724

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 {\text {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*}

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Mathematica [A]
time = 0.00, size = 31, normalized size = 1.00 \begin {gather*} \frac {\text {PolyLog}\left (1+n,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \end {gather*}

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 = 0.56, size = 32, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {\polylog \left (1+n , d \left (F^{c \left (b x +a \right )}\right )^{p}\right )}{b c p \ln \left (F \right )}\)	\(32\)
default	\(\frac {\polylog \left (1+n , d \left (F^{c \left (b x +a \right )}\right )^{p}\right )}{b c p \ln \left (F \right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {Li}_{n}\left (d \left (F^{c \left (a + b x\right )}\right )^{p}\right )\, dx \end {gather*}

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 = 0.00, 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(polylog(n,d*(F^(c*(b*x+a)))^p),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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