Integrand size = 9, antiderivative size = 71 \[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\frac {a q^2 x^{2+q} \operatorname {Hypergeometric2F1}\left (1,\frac {2+q}{q},2 \left (1+\frac {1}{q}\right ),a x^q\right )}{4 (2+q)}+\frac {1}{4} q x^2 \log \left (1-a x^q\right )+\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \] Output:
a*q^2*x^(2+q)*hypergeom([1, (2+q)/q],[2+2/q],a*x^q)/(8+4*q)+1/4*q*x^2*ln(1 -a*x^q)+1/2*x^2*polylog(2,a*x^q)
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97 \[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\frac {q x^2 \left (a q x^q \operatorname {Hypergeometric2F1}\left (1,\frac {2+q}{q},2+\frac {2}{q},a x^q\right )+(2+q) \log \left (1-a x^q\right )\right )}{4 (2+q)}+\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \] Input:
Integrate[x*PolyLog[2, a*x^q],x]
Output:
(q*x^2*(a*q*x^q*Hypergeometric2F1[1, (2 + q)/q, 2 + 2/q, a*x^q] + (2 + q)* Log[1 - a*x^q]))/(4*(2 + q)) + (x^2*PolyLog[2, a*x^q])/2
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, 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.
\(\displaystyle \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {PolyLog}\left (2,a x^q\right )-\frac {1}{2} q \int -x \log \left (1-a x^q\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} q \int x \log \left (1-a x^q\right )dx+\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,a x^q\right )\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {1}{2} q \left (\frac {1}{2} a q \int \frac {x^{q+1}}{1-a x^q}dx+\frac {1}{2} x^2 \log \left (1-a x^q\right )\right )+\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,a x^q\right )\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {1}{2} q \left (\frac {a q x^{q+2} \operatorname {Hypergeometric2F1}\left (1,\frac {q+2}{q},2 \left (1+\frac {1}{q}\right ),a x^q\right )}{2 (q+2)}+\frac {1}{2} x^2 \log \left (1-a x^q\right )\right )+\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,a x^q\right )\) |
Input:
Int[x*PolyLog[2, a*x^q],x]
Output:
(q*((a*q*x^(2 + q)*Hypergeometric2F1[1, (2 + q)/q, 2*(1 + q^(-1)), a*x^q]) /(2*(2 + q)) + (x^2*Log[1 - a*x^q])/2))/2 + (x^2*PolyLog[2, a*x^q])/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 = 0.68 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.52
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{-\frac {2}{q}} \left (-\frac {q^{2} x^{2} \left (-a \right )^{\frac {2}{q}} \ln \left (1-a \,x^{q}\right )}{4}-\frac {q \,x^{2} \left (-a \right )^{\frac {2}{q}} \left (1+\frac {q}{2}\right ) \operatorname {polylog}\left (2, a \,x^{q}\right )}{2+q}-\frac {q^{2} x^{2+q} a \left (-a \right )^{\frac {2}{q}} \operatorname {LerchPhi}\left (a \,x^{q}, 1, \frac {2+q}{q}\right )}{4}\right )}{q}\) | \(108\) |
Input:
int(x*polylog(2,a*x^q),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))
\[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { x {\rm Li}_2\left (a x^{q}\right ) \,d x } \] Input:
integrate(x*polylog(2,a*x^q),x, algorithm="fricas")
Output:
integral(x*dilog(a*x^q), x)
\[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int x \operatorname {Li}_{2}\left (a x^{q}\right )\, dx \] Input:
integrate(x*polylog(2,a*x**q),x)
Output:
Integral(x*polylog(2, a*x**q), x)
\[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { x {\rm Li}_2\left (a x^{q}\right ) \,d x } \] Input:
integrate(x*polylog(2,a*x^q),x, algorithm="maxima")
Output:
-1/8*q^2*x^2 + 1/4*q*x^2*log(-a*x^q + 1) + 1/2*x^2*dilog(a*x^q) - q^2*inte grate(1/4*x/(a*x^q - 1), x)
\[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { x {\rm Li}_2\left (a x^{q}\right ) \,d x } \] Input:
integrate(x*polylog(2,a*x^q),x, algorithm="giac")
Output:
integrate(x*dilog(a*x^q), x)
Timed out. \[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int x\,\mathrm {polylog}\left (2,a\,x^q\right ) \,d x \] Input:
int(x*polylog(2, a*x^q),x)
Output:
int(x*polylog(2, a*x^q), x)
\[ \int x \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int \mathit {polylog}\left (2, x^{q} a \right ) x d x \] Input:
int(x*polylog(2,a*x^q),x)
Output:
int(polylog(2,x**q*a)*x,x)