Integrand size = 11, antiderivative size = 78 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=-\frac {a q^2 x^{-2+q} \operatorname {Hypergeometric2F1}\left (1,-\frac {2-q}{q},2 \left (1-\frac {1}{q}\right ),a x^q\right )}{4 (2-q)}+\frac {q \log \left (1-a x^q\right )}{4 x^2}-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{2 x^2} \] Output:
-1/4*a*q^2*x^(-2+q)*hypergeom([1, -(2-q)/q],[2-2/q],a*x^q)/(2-q)+1/4*q*ln( 1-a*x^q)/x^2-1/2*polylog(2,a*x^q)/x^2
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=\frac {q \left (\frac {a q x^q \operatorname {Hypergeometric2F1}\left (1,\frac {-2+q}{q},2-\frac {2}{q},a x^q\right )}{-2+q}+\log \left (1-a x^q\right )\right )-2 \operatorname {PolyLog}\left (2,a x^q\right )}{4 x^2} \] Input:
Integrate[PolyLog[2, a*x^q]/x^3,x]
Output:
(q*((a*q*x^q*Hypergeometric2F1[1, (-2 + q)/q, 2 - 2/q, a*x^q])/(-2 + q) + Log[1 - a*x^q]) - 2*PolyLog[2, a*x^q])/(4*x^2)
Time = 0.43 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {7145, 25, 2905, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{2} q \int -\frac {\log \left (1-a x^q\right )}{x^3}dx-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} q \int \frac {\log \left (1-a x^q\right )}{x^3}dx-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{2 x^2}\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle -\frac {1}{2} q \left (-\frac {1}{2} a q \int \frac {x^{q-3}}{1-a x^q}dx-\frac {\log \left (1-a x^q\right )}{2 x^2}\right )-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{2 x^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {1}{2} q \left (\frac {a q x^{q-2} \operatorname {Hypergeometric2F1}\left (1,-\frac {2-q}{q},2 \left (1-\frac {1}{q}\right ),a x^q\right )}{2 (2-q)}-\frac {\log \left (1-a x^q\right )}{2 x^2}\right )-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{2 x^2}\) |
Input:
Int[PolyLog[2, a*x^q]/x^3,x]
Output:
-1/2*(q*((a*q*x^(-2 + q)*Hypergeometric2F1[1, -((2 - q)/q), 2*(1 - q^(-1)) , a*x^q])/(2*(2 - q)) - Log[1 - a*x^q]/(2*x^2))) - PolyLog[2, a*x^q]/(2*x^ 2)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ (m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Simp[b*e*n*(p/(f*(m + 1))) Int[x^(n - 1)*((f*x)^(m + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[p *(q/(m + 1)) Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 1.49 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.38
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{\frac {2}{q}} \left (-\frac {q^{2} \left (-a \right )^{-\frac {2}{q}} \ln \left (1-a \,x^{q}\right )}{4 x^{2}}-\frac {q \left (-a \right )^{-\frac {2}{q}} \left (1-\frac {q}{2}\right ) \operatorname {polylog}\left (2, a \,x^{q}\right )}{\left (-2+q \right ) x^{2}}-\frac {q^{2} x^{-2+q} a \left (-a \right )^{-\frac {2}{q}} \operatorname {LerchPhi}\left (a \,x^{q}, 1, \frac {-2+q}{q}\right )}{4}\right )}{q}\) | \(108\) |
Input:
int(polylog(2,a*x^q)/x^3,x,method=_RETURNVERBOSE)
Output:
-(-a)^(2/q)/q*(-1/4*q^2/x^2*(-a)^(-2/q)*ln(1-a*x^q)-q/(-2+q)/x^2*(-a)^(-2/ q)*(1-1/2*q)*polylog(2,a*x^q)-1/4*q^2*x^(-2+q)*a*(-a)^(-2/q)*LerchPhi(a*x^ q,1,(-2+q)/q))
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=\int { \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{3}} \,d x } \] Input:
integrate(polylog(2,a*x^q)/x^3,x, algorithm="fricas")
Output:
integral(dilog(a*x^q)/x^3, x)
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=\int \frac {\operatorname {Li}_{2}\left (a x^{q}\right )}{x^{3}}\, dx \] Input:
integrate(polylog(2,a*x**q)/x**3,x)
Output:
Integral(polylog(2, a*x**q)/x**3, x)
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=\int { \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{3}} \,d x } \] Input:
integrate(polylog(2,a*x^q)/x^3,x, algorithm="maxima")
Output:
-q^2*integrate(1/4/(a*x^3*x^q - x^3), x) + 1/8*(q^2 + 2*q*log(-a*x^q + 1) - 4*dilog(a*x^q))/x^2
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=\int { \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{3}} \,d x } \] Input:
integrate(polylog(2,a*x^q)/x^3,x, algorithm="giac")
Output:
integrate(dilog(a*x^q)/x^3, x)
Timed out. \[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=\int \frac {\mathrm {polylog}\left (2,a\,x^q\right )}{x^3} \,d x \] Input:
int(polylog(2, a*x^q)/x^3,x)
Output:
int(polylog(2, a*x^q)/x^3, x)
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^3} \, dx=\int \frac {\mathit {polylog}\left (2, x^{q} a \right )}{x^{3}}d x \] Input:
int(polylog(2,a*x^q)/x^3,x)
Output:
int(polylog(2,x**q*a)/x**3,x)