3.3.1 \(\int \frac {(a+b \log (c x^n)) \text {Li}_k(e x^q)}{x} \, dx\) [201]

Optimal. Leaf size=40 \[ \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {b n \text {Li}_{2+k}\left (e x^q\right )}{q^2} \]

[Out]

(a+b*ln(c*x^n))*polylog(1+k,e*x^q)/q-b*n*polylog(2+k,e*x^q)/q^2

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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2430, 6724} \begin {gather*} \frac {\text {PolyLog}\left (k+1,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{q}-\frac {b n \text {PolyLog}\left (k+2,e x^q\right )}{q^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])*PolyLog[k, e*x^q])/x,x]

[Out]

((a + b*Log[c*x^n])*PolyLog[1 + k, e*x^q])/q - (b*n*PolyLog[2 + k, e*x^q])/q^2

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

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 \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_k\left (e x^q\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {(b n) \int \frac {\text {Li}_{1+k}\left (e x^q\right )}{x} \, dx}{q}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {b n \text {Li}_{2+k}\left (e x^q\right )}{q^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.28 \begin {gather*} \frac {a \text {Li}_{1+k}\left (e x^q\right )}{q}+\frac {b \log \left (c x^n\right ) \text {Li}_{1+k}\left (e x^q\right )}{q}-\frac {b n \text {Li}_{2+k}\left (e x^q\right )}{q^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])*PolyLog[k, e*x^q])/x,x]

[Out]

(a*PolyLog[1 + k, e*x^q])/q + (b*Log[c*x^n]*PolyLog[1 + k, e*x^q])/q - (b*n*PolyLog[2 + k, e*x^q])/q^2

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (k , e \,x^{q}\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*polylog(k,e*x^q)/x,x)

[Out]

int((a+b*ln(c*x^n))*polylog(k,e*x^q)/x,x)

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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((a+b*log(c*x^n))*polylog(k,e*x^q)/x,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)*polylog(k, x^q*e)/x, 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((a+b*log(c*x^n))*polylog(k,e*x^q)/x,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*polylog(k, x^q*e)/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{k}\left (e x^{q}\right )}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*polylog(k,e*x**q)/x,x)

[Out]

Integral((a + b*log(c*x**n))*polylog(k, e*x**q)/x, 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((a+b*log(c*x^n))*polylog(k,e*x^q)/x,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*polylog(k, x^q*e)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {polylog}\left (k,e\,x^q\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((polylog(k, e*x^q)*(a + b*log(c*x^n)))/x,x)

[Out]

int((polylog(k, e*x^q)*(a + b*log(c*x^n)))/x, x)

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