3.19 \(\int x^5 \text{PolyLog}(2,a x^2) \, dx\)

Optimal. Leaf size=74 \[ \frac{1}{6} x^6 \text{PolyLog}\left (2,a x^2\right )-\frac{x^2}{18 a^2}-\frac{\log \left (1-a x^2\right )}{18 a^3}-\frac{x^4}{36 a}+\frac{1}{18} x^6 \log \left (1-a x^2\right )-\frac{x^6}{54} \]

[Out]

-x^2/(18*a^2) - x^4/(36*a) - x^6/54 - Log[1 - a*x^2]/(18*a^3) + (x^6*Log[1 - a*x^2])/18 + (x^6*PolyLog[2, a*x^
2])/6

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Rubi [A]  time = 0.0614215, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2454, 2395, 43} \[ \frac{1}{6} x^6 \text{PolyLog}\left (2,a x^2\right )-\frac{x^2}{18 a^2}-\frac{\log \left (1-a x^2\right )}{18 a^3}-\frac{x^4}{36 a}+\frac{1}{18} x^6 \log \left (1-a x^2\right )-\frac{x^6}{54} \]

Antiderivative was successfully verified.

[In]

Int[x^5*PolyLog[2, a*x^2],x]

[Out]

-x^2/(18*a^2) - x^4/(36*a) - x^6/54 - Log[1 - a*x^2]/(18*a^3) + (x^6*Log[1 - a*x^2])/18 + (x^6*PolyLog[2, a*x^
2])/6

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(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]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^5 \text{Li}_2\left (a x^2\right ) \, dx &=\frac{1}{6} x^6 \text{Li}_2\left (a x^2\right )+\frac{1}{3} \int x^5 \log \left (1-a x^2\right ) \, dx\\ &=\frac{1}{6} x^6 \text{Li}_2\left (a x^2\right )+\frac{1}{6} \operatorname{Subst}\left (\int x^2 \log (1-a x) \, dx,x,x^2\right )\\ &=\frac{1}{18} x^6 \log \left (1-a x^2\right )+\frac{1}{6} x^6 \text{Li}_2\left (a x^2\right )+\frac{1}{18} a \operatorname{Subst}\left (\int \frac{x^3}{1-a x} \, dx,x,x^2\right )\\ &=\frac{1}{18} x^6 \log \left (1-a x^2\right )+\frac{1}{6} x^6 \text{Li}_2\left (a x^2\right )+\frac{1}{18} a \operatorname{Subst}\left (\int \left (-\frac{1}{a^3}-\frac{x}{a^2}-\frac{x^2}{a}-\frac{1}{a^3 (-1+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{18 a^2}-\frac{x^4}{36 a}-\frac{x^6}{54}-\frac{\log \left (1-a x^2\right )}{18 a^3}+\frac{1}{18} x^6 \log \left (1-a x^2\right )+\frac{1}{6} x^6 \text{Li}_2\left (a x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0220162, size = 65, normalized size = 0.88 \[ \frac{18 a^3 x^6 \text{PolyLog}\left (2,a x^2\right )-a x^2 \left (2 a^2 x^4+3 a x^2+6\right )+6 \left (a^3 x^6-1\right ) \log \left (1-a x^2\right )}{108 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*PolyLog[2, a*x^2],x]

[Out]

(-(a*x^2*(6 + 3*a*x^2 + 2*a^2*x^4)) + 6*(-1 + a^3*x^6)*Log[1 - a*x^2] + 18*a^3*x^6*PolyLog[2, a*x^2])/(108*a^3
)

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Maple [A]  time = 0.044, size = 62, normalized size = 0.8 \begin{align*}{\frac{{x}^{6}{\it polylog} \left ( 2,a{x}^{2} \right ) }{6}}+{\frac{{x}^{6}\ln \left ( -a{x}^{2}+1 \right ) }{18}}-{\frac{{x}^{6}}{54}}-{\frac{{x}^{4}}{36\,a}}-{\frac{{x}^{2}}{18\,{a}^{2}}}-{\frac{\ln \left ( a{x}^{2}-1 \right ) }{18\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*polylog(2,a*x^2),x)

[Out]

1/6*x^6*polylog(2,a*x^2)+1/18*x^6*ln(-a*x^2+1)-1/54*x^6-1/36*x^4/a-1/18*x^2/a^2-1/18/a^3*ln(a*x^2-1)

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Maxima [A]  time = 1.00801, size = 84, normalized size = 1.14 \begin{align*} \frac{18 \, a^{3} x^{6}{\rm Li}_2\left (a x^{2}\right ) - 2 \, a^{3} x^{6} - 3 \, a^{2} x^{4} - 6 \, a x^{2} + 6 \,{\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right )}{108 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*polylog(2,a*x^2),x, algorithm="maxima")

[Out]

1/108*(18*a^3*x^6*dilog(a*x^2) - 2*a^3*x^6 - 3*a^2*x^4 - 6*a*x^2 + 6*(a^3*x^6 - 1)*log(-a*x^2 + 1))/a^3

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Fricas [A]  time = 2.5273, size = 142, normalized size = 1.92 \begin{align*} \frac{18 \, a^{3} x^{6}{\rm Li}_2\left (a x^{2}\right ) - 2 \, a^{3} x^{6} - 3 \, a^{2} x^{4} - 6 \, a x^{2} + 6 \,{\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right )}{108 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*polylog(2,a*x^2),x, algorithm="fricas")

[Out]

1/108*(18*a^3*x^6*dilog(a*x^2) - 2*a^3*x^6 - 3*a^2*x^4 - 6*a*x^2 + 6*(a^3*x^6 - 1)*log(-a*x^2 + 1))/a^3

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Sympy [A]  time = 42.5184, size = 56, normalized size = 0.76 \begin{align*} \begin{cases} - \frac{x^{6} \operatorname{Li}_{1}\left (a x^{2}\right )}{18} + \frac{x^{6} \operatorname{Li}_{2}\left (a x^{2}\right )}{6} - \frac{x^{6}}{54} - \frac{x^{4}}{36 a} - \frac{x^{2}}{18 a^{2}} + \frac{\operatorname{Li}_{1}\left (a x^{2}\right )}{18 a^{3}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*polylog(2,a*x**2),x)

[Out]

Piecewise((-x**6*polylog(1, a*x**2)/18 + x**6*polylog(2, a*x**2)/6 - x**6/54 - x**4/(36*a) - x**2/(18*a**2) +
polylog(1, a*x**2)/(18*a**3), Ne(a, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5}{\rm Li}_2\left (a x^{2}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*polylog(2,a*x^2),x, algorithm="giac")

[Out]

integrate(x^5*dilog(a*x^2), x)