3.69 \(\int \frac{\text{PolyLog}(3,a x)}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{4 \text{PolyLog}(2,a x)}{d \sqrt{d x}}-\frac{2 \text{PolyLog}(3,a x)}{d \sqrt{d x}}+\frac{16 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{8 \log (1-a x)}{d \sqrt{d x}} \]

[Out]

(16*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (8*Log[1 - a*x])/(d*Sqrt[d*x]) - (4*PolyLog[2, a*x
])/(d*Sqrt[d*x]) - (2*PolyLog[3, a*x])/(d*Sqrt[d*x])

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Rubi [A]  time = 0.0588706, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6591, 2395, 63, 206} \[ -\frac{4 \text{PolyLog}(2,a x)}{d \sqrt{d x}}-\frac{2 \text{PolyLog}(3,a x)}{d \sqrt{d x}}+\frac{16 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{8 \log (1-a x)}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x]/(d*x)^(3/2),x]

[Out]

(16*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (8*Log[1 - a*x])/(d*Sqrt[d*x]) - (4*PolyLog[2, a*x
])/(d*Sqrt[d*x]) - (2*PolyLog[3, a*x])/(d*Sqrt[d*x])

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 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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\text{Li}_3(a x)}{(d x)^{3/2}} \, dx &=-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}+2 \int \frac{\text{Li}_2(a x)}{(d x)^{3/2}} \, dx\\ &=-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}-4 \int \frac{\log (1-a x)}{(d x)^{3/2}} \, dx\\ &=\frac{8 \log (1-a x)}{d \sqrt{d x}}-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}+\frac{(8 a) \int \frac{1}{\sqrt{d x} (1-a x)} \, dx}{d}\\ &=\frac{8 \log (1-a x)}{d \sqrt{d x}}-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}+\frac{(16 a) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d^2}\\ &=\frac{16 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{8 \log (1-a x)}{d \sqrt{d x}}-\frac{4 \text{Li}_2(a x)}{d \sqrt{d x}}-\frac{2 \text{Li}_3(a x)}{d \sqrt{d x}}\\ \end{align*}

Mathematica [A]  time = 0.101918, size = 58, normalized size = 0.68 \[ \frac{2 x \left (-2 \text{PolyLog}(2,a x)-\text{PolyLog}(3,a x)+4 \log (1-a x)+8 \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x]/(d*x)^(3/2),x]

[Out]

(2*x*(8*Sqrt[a]*Sqrt[x]*ArcTanh[Sqrt[a]*Sqrt[x]] + 4*Log[1 - a*x] - 2*PolyLog[2, a*x] - PolyLog[3, a*x]))/(d*x
)^(3/2)

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Maple [A]  time = 0.056, size = 111, normalized size = 1.3 \begin{align*}{\frac{1}{a}{x}^{{\frac{3}{2}}} \left ( -a \right ) ^{{\frac{3}{2}}} \left ( -8\,{\frac{\sqrt{x}\sqrt{-a} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ) }{\sqrt{ax}}}+8\,{\frac{\sqrt{-a}\ln \left ( -ax+1 \right ) }{\sqrt{x}a}}-4\,{\frac{\sqrt{-a}{\it polylog} \left ( 2,ax \right ) }{\sqrt{x}a}}-2\,{\frac{\sqrt{-a}{\it polylog} \left ( 3,ax \right ) }{\sqrt{x}a}} \right ) \left ( dx \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x)/(d*x)^(3/2),x)

[Out]

1/(d*x)^(3/2)*x^(3/2)*(-a)^(3/2)/a*(-8*x^(1/2)*(-a)^(1/2)/(a*x)^(1/2)*(ln(1-(a*x)^(1/2))-ln(1+(a*x)^(1/2)))+8/
x^(1/2)*(-a)^(1/2)/a*ln(-a*x+1)-4/x^(1/2)*(-a)^(1/2)*polylog(2,a*x)/a-2/x^(1/2)*(-a)^(1/2)/a*polylog(3,a*x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.5892, size = 518, normalized size = 6.09 \begin{align*} \left [\frac{2 \,{\left (4 \, d x \sqrt{\frac{a}{d}} \log \left (\frac{a x + 2 \, \sqrt{d x} \sqrt{\frac{a}{d}} + 1}{a x - 1}\right ) - 2 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) + 4 \, \sqrt{d x} \log \left (-a x + 1\right ) - \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{d^{2} x}, -\frac{2 \,{\left (8 \, d x \sqrt{-\frac{a}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{a}{d}}}{a x}\right ) + 2 \, \sqrt{d x}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 4 \, \sqrt{d x} \log \left (-a x + 1\right ) + \sqrt{d x}{\rm polylog}\left (3, a x\right )\right )}}{d^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*(4*d*x*sqrt(a/d)*log((a*x + 2*sqrt(d*x)*sqrt(a/d) + 1)/(a*x - 1)) - 2*sqrt(d*x)*\%iint(a, x, -log(-a*x + 1)/
a, -log(-a*x + 1)/x) + 4*sqrt(d*x)*log(-a*x + 1) - sqrt(d*x)*polylog(3, a*x))/(d^2*x), -2*(8*d*x*sqrt(-a/d)*ar
ctan(sqrt(d*x)*sqrt(-a/d)/(a*x)) + 2*sqrt(d*x)*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) - 4*sqrt(d*x)*l
og(-a*x + 1) + sqrt(d*x)*polylog(3, a*x))/(d^2*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/(d*x)**(3/2),x)

[Out]

Integral(polylog(3, a*x)/(d*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x)}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(polylog(3, a*x)/(d*x)^(3/2), x)