3.27 \(\int x^2 \text {Li}_2(a x^2) \, dx\)

Optimal. Leaf size=63 \[ \frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{9 a^{3/2}}+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {2}{9} x^3 \log \left (1-a x^2\right )-\frac {4 x}{9 a}-\frac {4 x^3}{27} \]

[Out]

-4/9*x/a-4/27*x^3+4/9*arctanh(x*a^(1/2))/a^(3/2)+2/9*x^3*ln(-a*x^2+1)+1/3*x^3*polylog(2,a*x^2)

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Rubi [A]  time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6591, 2455, 302, 206} \[ \frac {1}{3} x^3 \text {PolyLog}\left (2,a x^2\right )+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{9 a^{3/2}}+\frac {2}{9} x^3 \log \left (1-a x^2\right )-\frac {4 x}{9 a}-\frac {4 x^3}{27} \]

Antiderivative was successfully verified.

[In]

Int[x^2*PolyLog[2, a*x^2],x]

[Out]

(-4*x)/(9*a) - (4*x^3)/27 + (4*ArcTanh[Sqrt[a]*x])/(9*a^(3/2)) + (2*x^3*Log[1 - a*x^2])/9 + (x^3*PolyLog[2, a*
x^2])/3

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])

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2455

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

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]

Rubi steps

\begin {align*} \int x^2 \text {Li}_2\left (a x^2\right ) \, dx &=\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {2}{3} \int x^2 \log \left (1-a x^2\right ) \, dx\\ &=\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {1}{9} (4 a) \int \frac {x^4}{1-a x^2} \, dx\\ &=\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {1}{9} (4 a) \int \left (-\frac {1}{a^2}-\frac {x^2}{a}+\frac {1}{a^2 \left (1-a x^2\right )}\right ) \, dx\\ &=-\frac {4 x}{9 a}-\frac {4 x^3}{27}+\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )+\frac {4 \int \frac {1}{1-a x^2} \, dx}{9 a}\\ &=-\frac {4 x}{9 a}-\frac {4 x^3}{27}+\frac {4 \tanh ^{-1}\left (\sqrt {a} x\right )}{9 a^{3/2}}+\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \text {Li}_2\left (a x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 57, normalized size = 0.90 \[ \frac {1}{27} \left (\frac {12 \tanh ^{-1}\left (\sqrt {a} x\right )}{a^{3/2}}+9 x^3 \text {Li}_2\left (a x^2\right )+6 x^3 \log \left (1-a x^2\right )-\frac {12 x}{a}-4 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*PolyLog[2, a*x^2],x]

[Out]

((-12*x)/a - 4*x^3 + (12*ArcTanh[Sqrt[a]*x])/a^(3/2) + 6*x^3*Log[1 - a*x^2] + 9*x^3*PolyLog[2, a*x^2])/27

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fricas [A]  time = 0.65, size = 143, normalized size = 2.27 \[ \left [\frac {9 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x + 6 \, \sqrt {a} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right )}{27 \, a^{2}}, \frac {9 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x - 12 \, \sqrt {-a} \arctan \left (\sqrt {-a} x\right )}{27 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/27*(9*a^2*x^3*dilog(a*x^2) + 6*a^2*x^3*log(-a*x^2 + 1) - 4*a^2*x^3 - 12*a*x + 6*sqrt(a)*log((a*x^2 + 2*sqrt
(a)*x + 1)/(a*x^2 - 1)))/a^2, 1/27*(9*a^2*x^3*dilog(a*x^2) + 6*a^2*x^3*log(-a*x^2 + 1) - 4*a^2*x^3 - 12*a*x -
12*sqrt(-a)*arctan(sqrt(-a)*x))/a^2]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm Li}_2\left (a x^{2}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*dilog(a*x^2), x)

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maple [A]  time = 0.01, size = 50, normalized size = 0.79 \[ -\frac {4 x}{9 a}-\frac {4 x^{3}}{27}+\frac {4 \arctanh \left (x \sqrt {a}\right )}{9 a^{\frac {3}{2}}}+\frac {2 x^{3} \ln \left (-a \,x^{2}+1\right )}{9}+\frac {x^{3} \polylog \left (2, a \,x^{2}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*x/a-4/27*x^3+4/9*arctanh(x*a^(1/2))/a^(3/2)+2/9*x^3*ln(-a*x^2+1)+1/3*x^3*polylog(2,a*x^2)

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maxima [A]  time = 0.42, size = 68, normalized size = 1.08 \[ \frac {9 \, a x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a x^{3} - 12 \, x}{27 \, a} - \frac {2 \, \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right )}{9 \, a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*a*x^3*dilog(a*x^2) + 6*a*x^3*log(-a*x^2 + 1) - 4*a*x^3 - 12*x)/a - 2/9*log((a*x - sqrt(a))/(a*x + sqrt
(a)))/a^(3/2)

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mupad [B]  time = 0.28, size = 52, normalized size = 0.83 \[ \frac {x^3\,\mathrm {polylog}\left (2,a\,x^2\right )}{3}-\frac {4\,x}{9\,a}+\frac {2\,x^3\,\ln \left (1-a\,x^2\right )}{9}-\frac {4\,x^3}{27}-\frac {\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{9\,a^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*polylog(2, a*x^2),x)

[Out]

(x^3*polylog(2, a*x^2))/3 - (atan(a^(1/2)*x*1i)*4i)/(9*a^(3/2)) - (4*x)/(9*a) + (2*x^3*log(1 - a*x^2))/9 - (4*
x^3)/27

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sympy [A]  time = 40.31, size = 83, normalized size = 1.32 \[ \begin {cases} - \frac {2 x^{3} \operatorname {Li}_{1}\left (a x^{2}\right )}{9} + \frac {x^{3} \operatorname {Li}_{2}\left (a x^{2}\right )}{3} - \frac {4 x^{3}}{27} - \frac {4 x}{9 a} - \frac {4 \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{9 a^{2} \sqrt {\frac {1}{a}}} - \frac {2 \operatorname {Li}_{1}\left (a x^{2}\right )}{9 a^{2} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*x**3*polylog(1, a*x**2)/9 + x**3*polylog(2, a*x**2)/3 - 4*x**3/27 - 4*x/(9*a) - 4*log(x - sqrt(1
/a))/(9*a**2*sqrt(1/a)) - 2*polylog(1, a*x**2)/(9*a**2*sqrt(1/a)), Ne(a, 0)), (0, True))

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