3.204 \(\int \frac {\text {Li}_k(e x^q)}{x (a+b \log (c x^n))^3} \, dx\)

Optimal. Leaf size=103 \[ \frac {q^2 \text {Int}\left (\frac {\text {Li}_{k-2}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )},x\right )}{2 b^2 n^2}-\frac {q \text {Li}_{k-1}\left (e x^q\right )}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])^3),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\text {Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx &=-\frac {\text {Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {q \int \frac {\text {Li}_{-1+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{2 b n}\\ &=-\frac {q \text {Li}_{-1+k}\left (e x^q\right )}{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Li}_k\left (e x^q\right )}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}+\frac {q^2 \int \frac {\text {Li}_{-2+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b^2 n^2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\text {Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])^3),x]

[Out]

Integrate[PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])^3), x]

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm polylog}\left (k, e x^{q}\right )}{b^{3} x \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b x \log \left (c x^{n}\right ) + a^{3} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(k,e*x^q)/x/(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

integral(polylog(k, e*x^q)/(b^3*x*log(c*x^n)^3 + 3*a*b^2*x*log(c*x^n)^2 + 3*a^2*b*x*log(c*x^n) + a^3*x), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(k,e*x^q)/x/(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

integrate(polylog(k, e*x^q)/((b*log(c*x^n) + a)^3*x), x)

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maple [A]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\polylog \left (k , e \,x^{q}\right )}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(k,e*x^q)/x/(b*ln(c*x^n)+a)^3,x)

[Out]

int(polylog(k,e*x^q)/x/(b*ln(c*x^n)+a)^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_{k}(e x^{q})}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(k,e*x^q)/x/(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

integrate(polylog(k, e*x^q)/((b*log(c*x^n) + a)^3*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {polylog}\left (k,e\,x^q\right )}{x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(k, e*x^q)/(x*(a + b*log(c*x^n))^3),x)

[Out]

int(polylog(k, e*x^q)/(x*(a + b*log(c*x^n))^3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{k}\left (e x^{q}\right )}{x \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(k,e*x**q)/x/(a+b*ln(c*x**n))**3,x)

[Out]

Integral(polylog(k, e*x**q)/(x*(a + b*log(c*x**n))**3), x)

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