3.3.3 \(\int \frac {\text {Li}_k(e x^q)}{x (a+b \log (c x^n))^2} \, dx\) [203]

Optimal. Leaf size=64 \[ -\frac {\text {Li}_k\left (e x^q\right )}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac {q \text {Int}\left (\frac {\text {Li}_{-1+k}\left (e x^q\right )}{x \left (a+b \log \left (c x^n\right )\right )},x\right )}{b n} \]

[Out]

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

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Rubi [A]
time = 0.05, 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 )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\polylog \left (k , e \,x^{q}\right )}{x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(polylog(k,e*x^q)/x/(a+b*ln(c*x^n))^2,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))^2,x, algorithm="maxima")

[Out]

integrate(polylog(k, x^q*e)/((b*log(c*x^n) + a)^2*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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Integral(polylog(k, e*x**q)/(x*(a + b*log(c*x**n))**2), 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))^2,x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {polylog}\left (k,e\,x^q\right )}{x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,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))^2),x)

[Out]

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

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