3.47 \(\int \text {Li}_2(a x^q) \, dx\)

Optimal. Leaf size=54 \[ \frac {a q^2 x^{q+1} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{q+1}+x \text {Li}_2\left (a x^q\right )+q x \log \left (1-a x^q\right ) \]

[Out]

a*q^2*x^(1+q)*hypergeom([1, 1+1/q],[2+1/q],a*x^q)/(1+q)+q*x*ln(1-a*x^q)+x*polylog(2,a*x^q)

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Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6586, 2448, 364} \[ x \text {PolyLog}\left (2,a x^q\right )+\frac {a q^2 x^{q+1} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{q+1}+q x \log \left (1-a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^q],x]

[Out]

(a*q^2*x^(1 + q)*Hypergeometric2F1[1, 1 + q^(-1), 2 + q^(-1), a*x^q])/(1 + q) + q*x*Log[1 - a*x^q] + x*PolyLog
[2, a*x^q]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 6586

Int[PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[x*PolyLog[n, a*(b*x^p)^q], x] - Dist[p*q, I
nt[PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, p, q}, x] && GtQ[n, 0]

Rubi steps

\begin {align*} \int \text {Li}_2\left (a x^q\right ) \, dx &=x \text {Li}_2\left (a x^q\right )+q \int \log \left (1-a x^q\right ) \, dx\\ &=q x \log \left (1-a x^q\right )+x \text {Li}_2\left (a x^q\right )+\left (a q^2\right ) \int \frac {x^q}{1-a x^q} \, dx\\ &=\frac {a q^2 x^{1+q} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{1+q}+q x \log \left (1-a x^q\right )+x \text {Li}_2\left (a x^q\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 51, normalized size = 0.94 \[ q x \left (\frac {a q x^q \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{q+1}+\log \left (1-a x^q\right )\right )+x \text {Li}_2\left (a x^q\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^q],x]

[Out]

q*x*((a*q*x^q*Hypergeometric2F1[1, 1 + q^(-1), 2 + q^(-1), a*x^q])/(1 + q) + Log[1 - a*x^q]) + x*PolyLog[2, a*
x^q]

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm Li}_2\left (a x^{q}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q),x, algorithm="fricas")

[Out]

integral(dilog(a*x^q), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Li}_2\left (a x^{q}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q),x, algorithm="giac")

[Out]

integrate(dilog(a*x^q), x)

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maple [C]  time = 0.14, size = 88, normalized size = 1.63 \[ -\frac {\left (-a \right )^{-\frac {1}{q}} \left (-q^{2} x \left (-a \right )^{\frac {1}{q}} \ln \left (1-a \,x^{q}\right )-q x \left (-a \right )^{\frac {1}{q}} \polylog \left (2, a \,x^{q}\right )-q^{2} x^{1+q} a \left (-a \right )^{\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {1+q}{q}\right )\right )}{q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^q),x)

[Out]

-1/q*(-a)^(-1/q)*(-q^2*x*(-a)^(1/q)*ln(1-a*x^q)-q*x*(-a)^(1/q)*polylog(2,a*x^q)-q^2*x^(1+q)*a*(-a)^(1/q)*Lerch
Phi(a*x^q,1,(1+q)/q))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -q^{2} x - q^{2} \int \frac {1}{a x^{q} - 1}\,{d x} + q x \log \left (-a x^{q} + 1\right ) + x {\rm Li}_2\left (a x^{q}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^q),x, algorithm="maxima")

[Out]

-q^2*x - q^2*integrate(1/(a*x^q - 1), x) + q*x*log(-a*x^q + 1) + x*dilog(a*x^q)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {polylog}\left (2,a\,x^q\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x^q),x)

[Out]

int(polylog(2, a*x^q), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {Li}_{2}\left (a x^{q}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x**q),x)

[Out]

Integral(polylog(2, a*x**q), x)

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