3.61 \(\int \frac{\text{PolyLog}(2,a x)}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=80 \[ \frac{2 \sqrt{d x} \text{PolyLog}(2,a x)}{d}+\frac{4 \sqrt{d x} \log (1-a x)}{d}+\frac{8 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}-\frac{8 \sqrt{d x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.0458829, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{2 \sqrt{d x} \text{PolyLog}(2,a x)}{d}+\frac{4 \sqrt{d x} \log (1-a x)}{d}+\frac{8 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}-\frac{8 \sqrt{d x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/Sqrt[d*x],x]

[Out]

(-8*Sqrt[d*x])/d + (8*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[a]*Sqrt[d]) + (4*Sqrt[d*x]*Log[1 - a*x])/d +
(2*Sqrt[d*x]*PolyLog[2, 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}_2(a x)}{\sqrt{d x}} \, dx &=\frac{2 \sqrt{d x} \text{Li}_2(a x)}{d}+2 \int \frac{\log (1-a x)}{\sqrt{d x}} \, dx\\ &=\frac{4 \sqrt{d x} \log (1-a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_2(a x)}{d}+\frac{(4 a) \int \frac{\sqrt{d x}}{1-a x} \, dx}{d}\\ &=-\frac{8 \sqrt{d x}}{d}+\frac{4 \sqrt{d x} \log (1-a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_2(a x)}{d}+4 \int \frac{1}{\sqrt{d x} (1-a x)} \, dx\\ &=-\frac{8 \sqrt{d x}}{d}+\frac{4 \sqrt{d x} \log (1-a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_2(a x)}{d}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{1-\frac{a x^2}{d}} \, dx,x,\sqrt{d x}\right )}{d}\\ &=-\frac{8 \sqrt{d x}}{d}+\frac{8 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt{a} \sqrt{d}}+\frac{4 \sqrt{d x} \log (1-a x)}{d}+\frac{2 \sqrt{d x} \text{Li}_2(a x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.078857, size = 63, normalized size = 0.79 \[ \frac{2 \sqrt{a} x \text{PolyLog}(2,a x)+4 \sqrt{a} x (\log (1-a x)-2)+8 \sqrt{x} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a} \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x]/Sqrt[d*x],x]

[Out]

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

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Maple [A]  time = 0.05, size = 69, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{dx}{\it polylog} \left ( 2,ax \right ) }{d}}+4\,{\frac{\sqrt{dx}}{d}\ln \left ({\frac{-adx+d}{d}} \right ) }-8\,{\frac{\sqrt{dx}}{d}}+8\,{\frac{1}{\sqrt{ad}}{\it Artanh} \left ({\frac{a\sqrt{dx}}{\sqrt{ad}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(2,a*x)/(d*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.61497, size = 331, normalized size = 4.14 \begin{align*} \left [\frac{2 \,{\left (\sqrt{d x}{\left (a{\rm Li}_2\left (a x\right ) + 2 \, a \log \left (-a x + 1\right ) - 4 \, a\right )} + 2 \, \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 (\sqrt{d x}{\left (a{\rm Li}_2\left (a x\right ) + 2 \, a \log \left (-a x + 1\right ) - 4 \, a\right )} - 4 \, \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(2,a*x)/(d*x)^(1/2),x, algorithm="fricas")

[Out]

[2*(sqrt(d*x)*(a*dilog(a*x) + 2*a*log(-a*x + 1) - 4*a) + 2*sqrt(a*d)*log((a*d*x + 2*sqrt(a*d)*sqrt(d*x) + d)/(
a*x - 1)))/(a*d), 2*(sqrt(d*x)*(a*dilog(a*x) + 2*a*log(-a*x + 1) - 4*a) - 4*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}_{2}\left (a x\right )}{\sqrt{d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(polylog(2, 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}_2\left (a x\right )}{\sqrt{d x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog(a*x)/sqrt(d*x), x)