3.21 \(\int x \text{PolyLog}(2,a x^2) \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{2} x^2 \text{PolyLog}\left (2,a x^2\right )-\frac{\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}-\frac{x^2}{2} \]

[Out]

-x^2/2 - ((1 - a*x^2)*Log[1 - a*x^2])/(2*a) + (x^2*PolyLog[2, a*x^2])/2

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Rubi [A]  time = 0.0252799, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6591, 2454, 2389, 2295} \[ \frac{1}{2} x^2 \text{PolyLog}\left (2,a x^2\right )-\frac{\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}-\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[x*PolyLog[2, a*x^2],x]

[Out]

-x^2/2 - ((1 - a*x^2)*Log[1 - a*x^2])/(2*a) + (x^2*PolyLog[2, a*x^2])/2

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 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin{align*} \int x \text{Li}_2\left (a x^2\right ) \, dx &=\frac{1}{2} x^2 \text{Li}_2\left (a x^2\right )+\int x \log \left (1-a x^2\right ) \, dx\\ &=\frac{1}{2} x^2 \text{Li}_2\left (a x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \log (1-a x) \, dx,x,x^2\right )\\ &=\frac{1}{2} x^2 \text{Li}_2\left (a x^2\right )-\frac{\operatorname{Subst}\left (\int \log (x) \, dx,x,1-a x^2\right )}{2 a}\\ &=-\frac{x^2}{2}-\frac{\left (1-a x^2\right ) \log \left (1-a x^2\right )}{2 a}+\frac{1}{2} x^2 \text{Li}_2\left (a x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0091547, size = 43, normalized size = 0.93 \[ \frac{a x^2 \text{PolyLog}\left (2,a x^2\right )-a x^2+\left (a x^2-1\right ) \log \left (1-a x^2\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x*PolyLog[2, a*x^2],x]

[Out]

(-(a*x^2) + (-1 + a*x^2)*Log[1 - a*x^2] + a*x^2*PolyLog[2, a*x^2])/(2*a)

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Maple [A]  time = 0.045, size = 52, normalized size = 1.1 \begin{align*}{\frac{{x}^{2}{\it polylog} \left ( 2,a{x}^{2} \right ) }{2}}+{\frac{\ln \left ( -a{x}^{2}+1 \right ){x}^{2}}{2}}-{\frac{{x}^{2}}{2}}-{\frac{\ln \left ( -a{x}^{2}+1 \right ) }{2\,a}}+{\frac{1}{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*polylog(2,a*x^2),x)

[Out]

1/2*x^2*polylog(2,a*x^2)+1/2*ln(-a*x^2+1)*x^2-1/2*x^2-1/2/a*ln(-a*x^2+1)+1/2/a

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Maxima [A]  time = 0.985418, size = 54, normalized size = 1.17 \begin{align*} \frac{a x^{2}{\rm Li}_2\left (a x^{2}\right ) - a x^{2} +{\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x^2*dilog(a*x^2) - a*x^2 + (a*x^2 - 1)*log(-a*x^2 + 1))/a

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Fricas [A]  time = 2.57043, size = 89, normalized size = 1.93 \begin{align*} \frac{a x^{2}{\rm Li}_2\left (a x^{2}\right ) - a x^{2} +{\left (a x^{2} - 1\right )} \log \left (-a x^{2} + 1\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*x^2*dilog(a*x^2) - a*x^2 + (a*x^2 - 1)*log(-a*x^2 + 1))/a

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Sympy [A]  time = 4.02697, size = 39, normalized size = 0.85 \begin{align*} \begin{cases} - \frac{x^{2} \operatorname{Li}_{1}\left (a x^{2}\right )}{2} + \frac{x^{2} \operatorname{Li}_{2}\left (a x^{2}\right )}{2} - \frac{x^{2}}{2} + \frac{\operatorname{Li}_{1}\left (a x^{2}\right )}{2 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**2*polylog(1, a*x**2)/2 + x**2*polylog(2, a*x**2)/2 - x**2/2 + polylog(1, a*x**2)/(2*a), Ne(a, 0
)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Li}_2\left (a x^{2}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*dilog(a*x^2), x)