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

Optimal. Leaf size=68 \[ -\frac{2 \text{PolyLog}(2,a x)}{d \sqrt{d x}}+\frac{8 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{4 \log (1-a x)}{d \sqrt{d x}} \]

[Out]

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

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Rubi [A]  time = 0.0431147, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6591, 2395, 63, 206} \[ -\frac{2 \text{PolyLog}(2,a x)}{d \sqrt{d x}}+\frac{8 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d x}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{4 \log (1-a x)}{d \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/(d*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0714846, size = 51, normalized size = 0.75 \[ \frac{2 x \left (-\text{PolyLog}(2,a x)+2 \log (1-a x)+4 \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{(d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.053, size = 63, normalized size = 0.9 \begin{align*} -2\,{\frac{{\it polylog} \left ( 2,ax \right ) }{d\sqrt{dx}}}+4\,{\frac{1}{d\sqrt{dx}}\ln \left ({\frac{-adx+d}{d}} \right ) }+8\,{\frac{a}{d\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)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.71198, size = 316, normalized size = 4.65 \begin{align*} \left [\frac{2 \,{\left (2 \, d x \sqrt{\frac{a}{d}} \log \left (\frac{a x + 2 \, \sqrt{d x} \sqrt{\frac{a}{d}} + 1}{a x - 1}\right ) - \sqrt{d x}{\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )}\right )}}{d^{2} x}, -\frac{2 \,{\left (4 \, d x \sqrt{-\frac{a}{d}} \arctan \left (\frac{\sqrt{d x} \sqrt{-\frac{a}{d}}}{a x}\right ) + \sqrt{d x}{\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )}\right )}}{d^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*(2*d*x*sqrt(a/d)*log((a*x + 2*sqrt(d*x)*sqrt(a/d) + 1)/(a*x - 1)) - sqrt(d*x)*(dilog(a*x) - 2*log(-a*x + 1)
))/(d^2*x), -2*(4*d*x*sqrt(-a/d)*arctan(sqrt(d*x)*sqrt(-a/d)/(a*x)) + sqrt(d*x)*(dilog(a*x) - 2*log(-a*x + 1))
)/(d^2*x)]

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

[Out]

Integral(polylog(2, 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}_2\left (a x\right )}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog(a*x)/(d*x)^(3/2), x)