3.1.39 \(\int x^4 \text {PolyLog}(3,a x^2) \, dx\) [39]

Optimal. Leaf size=87 \[ \frac {8 x}{125 a^2}+\frac {8 x^3}{375 a}+\frac {8 x^5}{625}-\frac {8 \tanh ^{-1}\left (\sqrt {a} x\right )}{125 a^{5/2}}-\frac {4}{125} x^5 \log \left (1-a x^2\right )-\frac {2}{25} x^5 \text {PolyLog}\left (2,a x^2\right )+\frac {1}{5} x^5 \text {PolyLog}\left (3,a x^2\right ) \]

[Out]

8/125*x/a^2+8/375*x^3/a+8/625*x^5-8/125*arctanh(x*a^(1/2))/a^(5/2)-4/125*x^5*ln(-a*x^2+1)-2/25*x^5*polylog(2,a
*x^2)+1/5*x^5*polylog(3,a*x^2)

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Rubi [A]
time = 0.04, antiderivative size = 87, 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, 2505, 308, 212} \begin {gather*} -\frac {8 \tanh ^{-1}\left (\sqrt {a} x\right )}{125 a^{5/2}}+\frac {8 x}{125 a^2}-\frac {2}{25} x^5 \text {Li}_2\left (a x^2\right )+\frac {1}{5} x^5 \text {Li}_3\left (a x^2\right )+\frac {8 x^3}{375 a}-\frac {4}{125} x^5 \log \left (1-a x^2\right )+\frac {8 x^5}{625} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*PolyLog[3, a*x^2],x]

[Out]

(8*x)/(125*a^2) + (8*x^3)/(375*a) + (8*x^5)/625 - (8*ArcTanh[Sqrt[a]*x])/(125*a^(5/2)) - (4*x^5*Log[1 - a*x^2]
)/125 - (2*x^5*PolyLog[2, a*x^2])/25 + (x^5*PolyLog[3, a*x^2])/5

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2505

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] - Dist[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]

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^4 \text {Li}_3\left (a x^2\right ) \, dx &=\frac {1}{5} x^5 \text {Li}_3\left (a x^2\right )-\frac {2}{5} \int x^4 \text {Li}_2\left (a x^2\right ) \, dx\\ &=-\frac {2}{25} x^5 \text {Li}_2\left (a x^2\right )+\frac {1}{5} x^5 \text {Li}_3\left (a x^2\right )-\frac {4}{25} \int x^4 \log \left (1-a x^2\right ) \, dx\\ &=-\frac {4}{125} x^5 \log \left (1-a x^2\right )-\frac {2}{25} x^5 \text {Li}_2\left (a x^2\right )+\frac {1}{5} x^5 \text {Li}_3\left (a x^2\right )-\frac {1}{125} (8 a) \int \frac {x^6}{1-a x^2} \, dx\\ &=-\frac {4}{125} x^5 \log \left (1-a x^2\right )-\frac {2}{25} x^5 \text {Li}_2\left (a x^2\right )+\frac {1}{5} x^5 \text {Li}_3\left (a x^2\right )-\frac {1}{125} (8 a) \int \left (-\frac {1}{a^3}-\frac {x^2}{a^2}-\frac {x^4}{a}+\frac {1}{a^3 \left (1-a x^2\right )}\right ) \, dx\\ &=\frac {8 x}{125 a^2}+\frac {8 x^3}{375 a}+\frac {8 x^5}{625}-\frac {4}{125} x^5 \log \left (1-a x^2\right )-\frac {2}{25} x^5 \text {Li}_2\left (a x^2\right )+\frac {1}{5} x^5 \text {Li}_3\left (a x^2\right )-\frac {8 \int \frac {1}{1-a x^2} \, dx}{125 a^2}\\ &=\frac {8 x}{125 a^2}+\frac {8 x^3}{375 a}+\frac {8 x^5}{625}-\frac {8 \tanh ^{-1}\left (\sqrt {a} x\right )}{125 a^{5/2}}-\frac {4}{125} x^5 \log \left (1-a x^2\right )-\frac {2}{25} x^5 \text {Li}_2\left (a x^2\right )+\frac {1}{5} x^5 \text {Li}_3\left (a x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 77, normalized size = 0.89 \begin {gather*} \frac {\frac {120 x}{a^2}+\frac {40 x^3}{a}+24 x^5-\frac {120 \tanh ^{-1}\left (\sqrt {a} x\right )}{a^{5/2}}-60 x^5 \log \left (1-a x^2\right )-150 x^5 \text {PolyLog}\left (2,a x^2\right )+375 x^5 \text {PolyLog}\left (3,a x^2\right )}{1875} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*PolyLog[3, a*x^2],x]

[Out]

((120*x)/a^2 + (40*x^3)/a + 24*x^5 - (120*ArcTanh[Sqrt[a]*x])/a^(5/2) - 60*x^5*Log[1 - a*x^2] - 150*x^5*PolyLo
g[2, a*x^2] + 375*x^5*PolyLog[3, a*x^2])/1875

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs. \(2(69)=138\).
time = 0.11, size = 144, normalized size = 1.66

method	result	size
meijerg	\(-\frac {\frac {2 x \left (-a \right )^{\frac {7}{2}} \left (168 a^{2} x^{4}+280 a \,x^{2}+840\right )}{13125 a^{3}}+\frac {8 x \left (-a \right )^{\frac {7}{2}} \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{125 a^{3} \sqrt {a \,x^{2}}}-\frac {8 x^{5} \left (-a \right )^{\frac {7}{2}} \ln \left (-a \,x^{2}+1\right )}{125 a}-\frac {4 x^{5} \left (-a \right )^{\frac {7}{2}} \polylog \left (2, a \,x^{2}\right )}{25 a}+\frac {2 x^{5} \left (-a \right )^{\frac {7}{2}} \polylog \left (3, a \,x^{2}\right )}{5 a}}{2 a^{2} \sqrt {-a}}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*polylog(3,a*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/2/a^2/(-a)^(1/2)*(2/13125*x*(-a)^(7/2)*(168*a^2*x^4+280*a*x^2+840)/a^3+8/125*x*(-a)^(7/2)/a^3/(a*x^2)^(1/2)
*(ln(1-(a*x^2)^(1/2))-ln(1+(a*x^2)^(1/2)))-8/125*x^5*(-a)^(7/2)/a*ln(-a*x^2+1)-4/25*x^5*(-a)^(7/2)/a*polylog(2
,a*x^2)+2/5*x^5*(-a)^(7/2)/a*polylog(3,a*x^2))

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Maxima [A]
time = 0.48, size = 95, normalized size = 1.09 \begin {gather*} -\frac {150 \, a^{2} x^{5} {\rm Li}_2\left (a x^{2}\right ) + 60 \, a^{2} x^{5} \log \left (-a x^{2} + 1\right ) - 375 \, a^{2} x^{5} {\rm Li}_{3}(a x^{2}) - 24 \, a^{2} x^{5} - 40 \, a x^{3} - 120 \, x}{1875 \, a^{2}} + \frac {4 \, \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right )}{125 \, a^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*polylog(3,a*x^2),x, algorithm="maxima")

[Out]

-1/1875*(150*a^2*x^5*dilog(a*x^2) + 60*a^2*x^5*log(-a*x^2 + 1) - 375*a^2*x^5*polylog(3, a*x^2) - 24*a^2*x^5 -
40*a*x^3 - 120*x)/a^2 + 4/125*log((a*x - sqrt(a))/(a*x + sqrt(a)))/a^(5/2)

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Fricas [A]
time = 0.69, size = 189, normalized size = 2.17 \begin {gather*} \left [-\frac {150 \, a^{3} x^{5} {\rm Li}_2\left (a x^{2}\right ) + 60 \, a^{3} x^{5} \log \left (-a x^{2} + 1\right ) - 375 \, a^{3} x^{5} {\rm polylog}\left (3, a x^{2}\right ) - 24 \, a^{3} x^{5} - 40 \, a^{2} x^{3} - 120 \, a x - 60 \, \sqrt {a} \log \left (\frac {a x^{2} - 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right )}{1875 \, a^{3}}, -\frac {150 \, a^{3} x^{5} {\rm Li}_2\left (a x^{2}\right ) + 60 \, a^{3} x^{5} \log \left (-a x^{2} + 1\right ) - 375 \, a^{3} x^{5} {\rm polylog}\left (3, a x^{2}\right ) - 24 \, a^{3} x^{5} - 40 \, a^{2} x^{3} - 120 \, a x - 120 \, \sqrt {-a} \arctan \left (\sqrt {-a} x\right )}{1875 \, a^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*polylog(3,a*x^2),x, algorithm="fricas")

[Out]

[-1/1875*(150*a^3*x^5*dilog(a*x^2) + 60*a^3*x^5*log(-a*x^2 + 1) - 375*a^3*x^5*polylog(3, a*x^2) - 24*a^3*x^5 -
40*a^2*x^3 - 120*a*x - 60*sqrt(a)*log((a*x^2 - 2*sqrt(a)*x + 1)/(a*x^2 - 1)))/a^3, -1/1875*(150*a^3*x^5*dilog
(a*x^2) + 60*a^3*x^5*log(-a*x^2 + 1) - 375*a^3*x^5*polylog(3, a*x^2) - 24*a^3*x^5 - 40*a^2*x^3 - 120*a*x - 120
*sqrt(-a)*arctan(sqrt(-a)*x))/a^3]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*polylog(3,a*x**2),x)

[Out]

Integral(x**4*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^4*polylog(3,a*x^2),x, algorithm="giac")

[Out]

integrate(x^4*polylog(3, a*x^2), x)

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Mupad [B]
time = 0.55, size = 72, normalized size = 0.83 \begin {gather*} \frac {x^5\,\mathrm {polylog}\left (3,a\,x^2\right )}{5}-\frac {2\,x^5\,\mathrm {polylog}\left (2,a\,x^2\right )}{25}+\frac {8\,x}{125\,a^2}-\frac {4\,x^5\,\ln \left (1-a\,x^2\right )}{125}+\frac {8\,x^5}{625}+\frac {8\,x^3}{375\,a}+\frac {\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{125\,a^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*polylog(3, a*x^2),x)

[Out]

(atan(a^(1/2)*x*1i)*8i)/(125*a^(5/2)) - (2*x^5*polylog(2, a*x^2))/25 + (x^5*polylog(3, a*x^2))/5 + (8*x)/(125*
a^2) - (4*x^5*log(1 - a*x^2))/125 + (8*x^5)/625 + (8*x^3)/(375*a)

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