3.1.32 \(\int x^5 \text {PolyLog}(3,a x^2) \, dx\) [32]

Optimal. Leaf size=88 \[ \frac {x^2}{54 a^2}+\frac {x^4}{108 a}+\frac {x^6}{162}+\frac {\log \left (1-a x^2\right )}{54 a^3}-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {PolyLog}\left (2,a x^2\right )+\frac {1}{6} x^6 \text {PolyLog}\left (3,a x^2\right ) \]

[Out]

1/54*x^2/a^2+1/108*x^4/a+1/162*x^6+1/54*ln(-a*x^2+1)/a^3-1/54*x^6*ln(-a*x^2+1)-1/18*x^6*polylog(2,a*x^2)+1/6*x
^6*polylog(3,a*x^2)

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Rubi [A]
time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6726, 2504, 2442, 45} \begin {gather*} \frac {\log \left (1-a x^2\right )}{54 a^3}+\frac {x^2}{54 a^2}-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )+\frac {x^4}{108 a}-\frac {1}{54} x^6 \log \left (1-a x^2\right )+\frac {x^6}{162} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*PolyLog[3, a*x^2],x]

[Out]

x^2/(54*a^2) + x^4/(108*a) + x^6/162 + Log[1 - a*x^2]/(54*a^3) - (x^6*Log[1 - a*x^2])/54 - (x^6*PolyLog[2, a*x
^2])/18 + (x^6*PolyLog[3, a*x^2])/6

Rule 45

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])

Rule 2442

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 2504

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 6726

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]

Rubi steps

\begin {align*} \int x^5 \text {Li}_3\left (a x^2\right ) \, dx &=\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{3} \int x^5 \text {Li}_2\left (a x^2\right ) \, dx\\ &=-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{9} \int x^5 \log \left (1-a x^2\right ) \, dx\\ &=-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{18} \text {Subst}\left (\int x^2 \log (1-a x) \, dx,x,x^2\right )\\ &=-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{54} a \text {Subst}\left (\int \frac {x^3}{1-a x} \, dx,x,x^2\right )\\ &=-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )-\frac {1}{54} a \text {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}{54 a^2}+\frac {x^4}{108 a}+\frac {x^6}{162}+\frac {\log \left (1-a x^2\right )}{54 a^3}-\frac {1}{54} x^6 \log \left (1-a x^2\right )-\frac {1}{18} x^6 \text {Li}_2\left (a x^2\right )+\frac {1}{6} x^6 \text {Li}_3\left (a x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 88, normalized size = 1.00 \begin {gather*} \frac {6 a x^2+3 a^2 x^4+2 a^3 x^6+6 \log \left (1-a x^2\right )-6 a^3 x^6 \log \left (1-a x^2\right )-18 a^3 x^6 \text {PolyLog}\left (2,a x^2\right )+54 a^3 x^6 \text {PolyLog}\left (3,a x^2\right )}{324 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*PolyLog[3, a*x^2],x]

[Out]

(6*a*x^2 + 3*a^2*x^4 + 2*a^3*x^6 + 6*Log[1 - a*x^2] - 6*a^3*x^6*Log[1 - a*x^2] - 18*a^3*x^6*PolyLog[2, a*x^2]
+ 54*a^3*x^6*PolyLog[3, a*x^2])/(324*a^3)

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Maple [A]
time = 0.04, size = 80, normalized size = 0.91

method	result	size
meijerg	\(\frac {\frac {x^{2} a \left (4 a^{2} x^{4}+6 a \,x^{2}+12\right )}{324}+\frac {\left (-4 a^{3} x^{6}+4\right ) \ln \left (-a \,x^{2}+1\right )}{108}-\frac {x^{6} a^{3} \polylog \left (2, a \,x^{2}\right )}{9}+\frac {x^{6} a^{3} \polylog \left (3, a \,x^{2}\right )}{3}}{2 a^{3}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*polylog(3,a*x^2),x,method=_RETURNVERBOSE)

[Out]

1/2/a^3*(1/324*x^2*a*(4*a^2*x^4+6*a*x^2+12)+1/108*(-4*a^3*x^6+4)*ln(-a*x^2+1)-1/9*x^6*a^3*polylog(2,a*x^2)+1/3
*x^6*a^3*polylog(3,a*x^2))

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Maxima [A]
time = 0.27, size = 77, normalized size = 0.88 \begin {gather*} -\frac {18 \, a^{3} x^{6} {\rm Li}_2\left (a x^{2}\right ) - 54 \, a^{3} x^{6} {\rm Li}_{3}(a x^{2}) - 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 )}{324 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*polylog(3,a*x^2),x, algorithm="maxima")

[Out]

-1/324*(18*a^3*x^6*dilog(a*x^2) - 54*a^3*x^6*polylog(3, 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 = 0.43, size = 77, normalized size = 0.88 \begin {gather*} -\frac {18 \, a^{3} x^{6} {\rm Li}_2\left (a x^{2}\right ) - 54 \, a^{3} x^{6} {\rm polylog}\left (3, 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 )}{324 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*polylog(3,a*x^2),x, algorithm="fricas")

[Out]

-1/324*(18*a^3*x^6*dilog(a*x^2) - 54*a^3*x^6*polylog(3, 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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*polylog(3,a*x**2),x)

[Out]

Integral(x**5*polylog(3, a*x**2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*polylog(3,a*x^2),x, algorithm="giac")

[Out]

integrate(x^5*polylog(3, a*x^2), x)

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Mupad [B]
time = 0.34, size = 73, normalized size = 0.83 \begin {gather*} \frac {x^6\,\mathrm {polylog}\left (3,a\,x^2\right )}{6}-\frac {x^6\,\mathrm {polylog}\left (2,a\,x^2\right )}{18}+\frac {\ln \left (a\,x^2-1\right )}{54\,a^3}-\frac {x^6\,\ln \left (1-a\,x^2\right )}{54}+\frac {x^6}{162}+\frac {x^2}{54\,a^2}+\frac {x^4}{108\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*polylog(3, a*x^2),x)

[Out]

(x^6*polylog(3, a*x^2))/6 - (x^6*polylog(2, a*x^2))/18 + log(a*x^2 - 1)/(54*a^3) - (x^6*log(1 - a*x^2))/54 + x
^6/162 + x^2/(54*a^2) + x^4/(108*a)

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