Integrand size = 11, antiderivative size = 71 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\frac {a q^2 x^{3+q} \operatorname {Hypergeometric2F1}\left (1,\frac {3+q}{q},2+\frac {3}{q},a x^q\right )}{9 (3+q)}+\frac {1}{9} q x^3 \log \left (1-a x^q\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^q\right ) \] Output:
a*q^2*x^(3+q)*hypergeom([1, (3+q)/q],[2+3/q],a*x^q)/(27+9*q)+1/9*q*x^3*ln( 1-a*x^q)+1/3*x^3*polylog(2,a*x^q)
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\frac {q x^3 \left (a q x^q \operatorname {Hypergeometric2F1}\left (1,\frac {3+q}{q},2+\frac {3}{q},a x^q\right )+(3+q) \log \left (1-a x^q\right )\right )}{9 (3+q)}+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^q\right ) \] Input:
Integrate[x^2*PolyLog[2, a*x^q],x]
Output:
(q*x^3*(a*q*x^q*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q] + (3 + q)* Log[1 - a*x^q]))/(9*(3 + q)) + (x^3*PolyLog[2, a*x^q])/3
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, 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.
\(\displaystyle \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^q\right )-\frac {1}{3} q \int -x^2 \log \left (1-a x^q\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} q \int x^2 \log \left (1-a x^q\right )dx+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^q\right )\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {1}{3} q \left (\frac {1}{3} a q \int \frac {x^{q+2}}{1-a x^q}dx+\frac {1}{3} x^3 \log \left (1-a x^q\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^q\right )\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {1}{3} q \left (\frac {a q x^{q+3} \operatorname {Hypergeometric2F1}\left (1,\frac {q+3}{q},2+\frac {3}{q},a x^q\right )}{3 (q+3)}+\frac {1}{3} x^3 \log \left (1-a x^q\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^q\right )\) |
Input:
Int[x^2*PolyLog[2, a*x^q],x]
Output:
(q*((a*q*x^(3 + q)*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q])/(3*(3 + q)) + (x^3*Log[1 - a*x^q])/3))/3 + (x^3*PolyLog[2, a*x^q])/3
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 = 1.41 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.52
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{-\frac {3}{q}} \left (-\frac {q^{2} x^{3} \left (-a \right )^{\frac {3}{q}} \ln \left (1-a \,x^{q}\right )}{9}-\frac {q \,x^{3} \left (-a \right )^{\frac {3}{q}} \left (1+\frac {q}{3}\right ) \operatorname {polylog}\left (2, a \,x^{q}\right )}{3+q}-\frac {q^{2} x^{3+q} a \left (-a \right )^{\frac {3}{q}} \operatorname {LerchPhi}\left (a \,x^{q}, 1, \frac {3+q}{q}\right )}{9}\right )}{q}\) | \(108\) |
Input:
int(x^2*polylog(2,a*x^q),x,method=_RETURNVERBOSE)
Output:
-(-a)^(-3/q)/q*(-1/9*q^2*x^3*(-a)^(3/q)*ln(1-a*x^q)-q/(3+q)*x^3*(-a)^(3/q) *(1+1/3*q)*polylog(2,a*x^q)-1/9*q^2*x^(3+q)*a*(-a)^(3/q)*LerchPhi(a*x^q,1, (3+q)/q))
\[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { x^{2} {\rm Li}_2\left (a x^{q}\right ) \,d x } \] Input:
integrate(x^2*polylog(2,a*x^q),x, algorithm="fricas")
Output:
integral(x^2*dilog(a*x^q), x)
\[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int x^{2} \operatorname {Li}_{2}\left (a x^{q}\right )\, dx \] Input:
integrate(x**2*polylog(2,a*x**q),x)
Output:
Integral(x**2*polylog(2, a*x**q), x)
\[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { x^{2} {\rm Li}_2\left (a x^{q}\right ) \,d x } \] Input:
integrate(x^2*polylog(2,a*x^q),x, algorithm="maxima")
Output:
-1/27*q^2*x^3 + 1/9*q*x^3*log(-a*x^q + 1) + 1/3*x^3*dilog(a*x^q) - q^2*int egrate(1/9*x^2/(a*x^q - 1), x)
\[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int { x^{2} {\rm Li}_2\left (a x^{q}\right ) \,d x } \] Input:
integrate(x^2*polylog(2,a*x^q),x, algorithm="giac")
Output:
integrate(x^2*dilog(a*x^q), x)
Timed out. \[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int x^2\,\mathrm {polylog}\left (2,a\,x^q\right ) \,d x \] Input:
int(x^2*polylog(2, a*x^q),x)
Output:
int(x^2*polylog(2, a*x^q), x)
\[ \int x^2 \operatorname {PolyLog}\left (2,a x^q\right ) \, dx=\int \mathit {polylog}\left (2, x^{q} a \right ) x^{2}d x \] Input:
int(x^2*polylog(2,a*x^q),x)
Output:
int(polylog(2,x**q*a)*x**2,x)