3.3.4 \(\int \frac {\text {Li}_k(e x^q)}{x (a+b \log (c x^n))^3} \, dx\) [204]

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

[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 = 0.08, 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 = {} \begin {gather*} \int \frac {\text {PolyLog}\left (k,e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/2*(q*PolyLog[-1 + k, e*x^q])/(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 = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {Li}_k\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx \end {gather*}

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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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\polylog \left (k , e \,x^{q}\right )}{x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
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(k,e*x^q)/x/(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
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(k,e*x^q)/x/(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

integral(polylog(k, x^q*e)/(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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{k}\left (e x^{q}\right )}{x \left (a + b \log {\left (c x^{n} \right )}\right )^{3}}\, dx \end {gather*}

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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Giac [A]
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(k,e*x^q)/x/(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

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

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

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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