Integrand size = 11, antiderivative size = 69 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=-\frac {a q^2 x^{-1+q} \operatorname {Hypergeometric2F1}\left (1,-\frac {1-q}{q},2-\frac {1}{q},a x^q\right )}{1-q}+\frac {q \log \left (1-a x^q\right )}{x}-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x} \] Output:
-a*q^2*x^(-1+q)*hypergeom([1, -(1-q)/q],[2-1/q],a*x^q)/(1-q)+q*ln(1-a*x^q) /x-polylog(2,a*x^q)/x
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=\frac {q \left (\frac {a q x^q \operatorname {Hypergeometric2F1}\left (1,\frac {-1+q}{q},2-\frac {1}{q},a x^q\right )}{-1+q}+\log \left (1-a x^q\right )\right )}{x}-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x} \] Input:
Integrate[PolyLog[2, a*x^q]/x^2,x]
Output:
(q*((a*q*x^q*Hypergeometric2F1[1, (-1 + q)/q, 2 - q^(-1), a*x^q])/(-1 + q) + Log[1 - a*x^q]))/x - PolyLog[2, a*x^q]/x
Time = 0.42 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01, 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.
\(\displaystyle \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle q \int -\frac {\log \left (1-a x^q\right )}{x^2}dx-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -q \int \frac {\log \left (1-a x^q\right )}{x^2}dx-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x}\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle -q \left (-a q \int \frac {x^{q-2}}{1-a x^q}dx-\frac {\log \left (1-a x^q\right )}{x}\right )-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -q \left (\frac {a q x^{q-1} \operatorname {Hypergeometric2F1}\left (1,-\frac {1-q}{q},2-\frac {1}{q},a x^q\right )}{1-q}-\frac {\log \left (1-a x^q\right )}{x}\right )-\frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x}\) |
Input:
Int[PolyLog[2, a*x^q]/x^2,x]
Output:
-(q*((a*q*x^(-1 + q)*Hypergeometric2F1[1, -((1 - q)/q), 2 - q^(-1), a*x^q] )/(1 - q) - Log[1 - a*x^q]/x)) - PolyLog[2, a*x^q]/x
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 = 0.81 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.54
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{\frac {1}{q}} \left (-\frac {q^{2} \left (-a \right )^{-\frac {1}{q}} \ln \left (1-a \,x^{q}\right )}{x}-\frac {q \left (-a \right )^{-\frac {1}{q}} \left (1-q \right ) \operatorname {polylog}\left (2, a \,x^{q}\right )}{\left (-1+q \right ) x}-q^{2} x^{-1+q} a \left (-a \right )^{-\frac {1}{q}} \operatorname {LerchPhi}\left (a \,x^{q}, 1, \frac {-1+q}{q}\right )\right )}{q}\) | \(106\) |
Input:
int(polylog(2,a*x^q)/x^2,x,method=_RETURNVERBOSE)
Output:
-(-a)^(1/q)/q*(-q^2/x*(-a)^(-1/q)*ln(1-a*x^q)-q/(-1+q)/x*(-a)^(-1/q)*(1-q) *polylog(2,a*x^q)-q^2*x^(-1+q)*a*(-a)^(-1/q)*LerchPhi(a*x^q,1,(-1+q)/q))
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=\int { \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{2}} \,d x } \] Input:
integrate(polylog(2,a*x^q)/x^2,x, algorithm="fricas")
Output:
integral(dilog(a*x^q)/x^2, x)
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=\int \frac {\operatorname {Li}_{2}\left (a x^{q}\right )}{x^{2}}\, dx \] Input:
integrate(polylog(2,a*x**q)/x**2,x)
Output:
Integral(polylog(2, a*x**q)/x**2, x)
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=\int { \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{2}} \,d x } \] Input:
integrate(polylog(2,a*x^q)/x^2,x, algorithm="maxima")
Output:
-q^2*integrate(1/(a*x^2*x^q - x^2), x) + (q^2 + q*log(-a*x^q + 1) - dilog( a*x^q))/x
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=\int { \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{2}} \,d x } \] Input:
integrate(polylog(2,a*x^q)/x^2,x, algorithm="giac")
Output:
integrate(dilog(a*x^q)/x^2, x)
Timed out. \[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=\int \frac {\mathrm {polylog}\left (2,a\,x^q\right )}{x^2} \,d x \] Input:
int(polylog(2, a*x^q)/x^2,x)
Output:
int(polylog(2, a*x^q)/x^2, x)
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^q\right )}{x^2} \, dx=\int \frac {\mathit {polylog}\left (2, x^{q} a \right )}{x^{2}}d x \] Input:
int(polylog(2,a*x^q)/x^2,x)
Output:
int(polylog(2,x**q*a)/x**2,x)