3.68 \(\int \frac{\text{PolyLog}(3,a x)}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{4 \sqrt{d x} \text{PolyLog}(2,a x)}{d}+\frac{2 \sqrt{d x} \text{PolyLog}(3,a x)}{d}-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{16 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}+\frac{16 \sqrt{d x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.0621491, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6591, 2395, 50, 63, 206} \[ -\frac{4 \sqrt{d x} \text{PolyLog}(2,a x)}{d}+\frac{2 \sqrt{d x} \text{PolyLog}(3,a x)}{d}-\frac{8 \sqrt{d x} \log (1-a x)}{d}-\frac{16 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}+\frac{16 \sqrt{d x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x]/Sqrt[d*x],x]

[Out]

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

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

Mathematica [A]  time = 0.138969, size = 57, normalized size = 0.59 \[ \frac{2 x \left (-2 \text{PolyLog}(2,a x)+\text{PolyLog}(3,a x)-4 \log (1-a x)-\frac{8 \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a} \sqrt{x}}+8\right )}{\sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x]/Sqrt[d*x],x]

[Out]

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

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Maple [A]  time = 0.057, size = 127, normalized size = 1.3 \begin{align*}{\frac{1}{a}\sqrt{x}\sqrt{-a} \left ( 16\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}}{a}}+8\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2} \left ( \ln \left ( 1-\sqrt{ax} \right ) -\ln \left ( 1+\sqrt{ax} \right ) \right ) }{a\sqrt{ax}}}-8\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}\ln \left ( -ax+1 \right ) }{a}}-4\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}{\it polylog} \left ( 2,ax \right ) }{a}}+2\,{\frac{\sqrt{x} \left ( -a \right ) ^{3/2}{\it polylog} \left ( 3,ax \right ) }{a}} \right ){\frac{1}{\sqrt{dx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x)/(d*x)^(1/2),x)

[Out]

1/(d*x)^(1/2)*x^(1/2)*(-a)^(1/2)/a*(16*x^(1/2)*(-a)^(3/2)/a+8*x^(1/2)*(-a)^(3/2)/a/(a*x)^(1/2)*(ln(1-(a*x)^(1/
2))-ln(1+(a*x)^(1/2)))-8*x^(1/2)*(-a)^(3/2)/a*ln(-a*x+1)-4*x^(1/2)*(-a)^(3/2)*polylog(2,a*x)/a+2*x^(1/2)*(-a)^
(3/2)/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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.72649, size = 547, normalized size = 5.64 \begin{align*} \left [-\frac{2 \,{\left (2 \, \sqrt{d x} a{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - \sqrt{d x} a{\rm polylog}\left (3, a x\right ) + 4 \, \sqrt{d x}{\left (a \log \left (-a x + 1\right ) - 2 \, a\right )} - 4 \, \sqrt{a d} \log \left (\frac{a d x - 2 \, \sqrt{a d} \sqrt{d x} + d}{a x - 1}\right )\right )}}{a d}, -\frac{2 \,{\left (2 \, \sqrt{d x} a{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - \sqrt{d x} a{\rm polylog}\left (3, a x\right ) + 4 \, \sqrt{d x}{\left (a \log \left (-a x + 1\right ) - 2 \, a\right )} - 8 \, \sqrt{-a d} \arctan \left (\frac{\sqrt{-a d} \sqrt{d x}}{a d x}\right )\right )}}{a d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/(d*x)^(1/2),x, algorithm="fricas")

[Out]

[-2*(2*sqrt(d*x)*a*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) - sqrt(d*x)*a*polylog(3, a*x) + 4*sqrt(d*x)
*(a*log(-a*x + 1) - 2*a) - 4*sqrt(a*d)*log((a*d*x - 2*sqrt(a*d)*sqrt(d*x) + d)/(a*x - 1)))/(a*d), -2*(2*sqrt(d
*x)*a*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x + 1)/x) - sqrt(d*x)*a*polylog(3, a*x) + 4*sqrt(d*x)*(a*log(-a*x
+ 1) - 2*a) - 8*sqrt(-a*d)*arctan(sqrt(-a*d)*sqrt(d*x)/(a*d*x)))/(a*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/(d*x)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/(d*x)^(1/2),x, algorithm="giac")

[Out]

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