3.73 \(\int \sqrt{d x} \text{PolyLog}(2,a x^2) \, dx\)

Optimal. Leaf size=125 \[ \frac{2 (d x)^{3/2} \text{PolyLog}\left (2,a x^2\right )}{3 d}-\frac{16 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 a^{3/4}}+\frac{16 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 a^{3/4}}+\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}-\frac{32 (d x)^{3/2}}{27 d} \]

[Out]

(-32*(d*x)^(3/2))/(27*d) - (16*Sqrt[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*a^(3/4)) + (16*Sqrt[d]*ArcTanh[
(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*a^(3/4)) + (8*(d*x)^(3/2)*Log[1 - a*x^2])/(9*d) + (2*(d*x)^(3/2)*PolyLog[2, a
*x^2])/(3*d)

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Rubi [A]  time = 0.086485, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {6591, 2455, 16, 321, 329, 298, 205, 208} \[ \frac{2 (d x)^{3/2} \text{PolyLog}\left (2,a x^2\right )}{3 d}-\frac{16 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 a^{3/4}}+\frac{16 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 a^{3/4}}+\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}-\frac{32 (d x)^{3/2}}{27 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*PolyLog[2, a*x^2],x]

[Out]

(-32*(d*x)^(3/2))/(27*d) - (16*Sqrt[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*a^(3/4)) + (16*Sqrt[d]*ArcTanh[
(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*a^(3/4)) + (8*(d*x)^(3/2)*Log[1 - a*x^2])/(9*d) + (2*(d*x)^(3/2)*PolyLog[2, a
*x^2])/(3*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 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 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \sqrt{d x} \text{Li}_2\left (a x^2\right ) \, dx &=\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{3 d}+\frac{4}{3} \int \sqrt{d x} \log \left (1-a x^2\right ) \, dx\\ &=\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{3 d}+\frac{(16 a) \int \frac{x (d x)^{3/2}}{1-a x^2} \, dx}{9 d}\\ &=\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{3 d}+\frac{(16 a) \int \frac{(d x)^{5/2}}{1-a x^2} \, dx}{9 d^2}\\ &=-\frac{32 (d x)^{3/2}}{27 d}+\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{3 d}+\frac{16}{9} \int \frac{\sqrt{d x}}{1-a x^2} \, dx\\ &=-\frac{32 (d x)^{3/2}}{27 d}+\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{3 d}+\frac{32 \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{a x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{9 d}\\ &=-\frac{32 (d x)^{3/2}}{27 d}+\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{3 d}+\frac{(16 d) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{9 \sqrt{a}}-\frac{(16 d) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{a} x^2} \, dx,x,\sqrt{d x}\right )}{9 \sqrt{a}}\\ &=-\frac{32 (d x)^{3/2}}{27 d}-\frac{16 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 a^{3/4}}+\frac{16 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{d x}}{\sqrt{d}}\right )}{9 a^{3/4}}+\frac{8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac{2 (d x)^{3/2} \text{Li}_2\left (a x^2\right )}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0696545, size = 91, normalized size = 0.73 \[ \frac{2 \sqrt{d x} \left (9 x^{3/2} \text{PolyLog}\left (2,a x^2\right )+\frac{4 \left (a^{3/4} x^{3/2} \left (3 \log \left (1-a x^2\right )-4\right )-6 \tan ^{-1}\left (\sqrt [4]{a} \sqrt{x}\right )+6 \tanh ^{-1}\left (\sqrt [4]{a} \sqrt{x}\right )\right )}{a^{3/4}}\right )}{27 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.052, size = 139, normalized size = 1.1 \begin{align*}{\frac{2\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{3\,d} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{8}{9\,d} \left ( dx \right ) ^{{\frac{3}{2}}}\ln \left ({\frac{-a{d}^{2}{x}^{2}+{d}^{2}}{{d}^{2}}} \right ) }-{\frac{32}{27\,d} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{16\,d}{9\,a}\arctan \left ({\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{a}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{a}}}}}}+{\frac{8\,d}{9\,a}\ln \left ({ \left ( \sqrt{dx}+\sqrt [4]{{\frac{{d}^{2}}{a}}} \right ) \left ( \sqrt{dx}-\sqrt [4]{{\frac{{d}^{2}}{a}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(d*x)^(3/2)*polylog(2,a*x^2)/d+8/9/d*(d*x)^(3/2)*ln((-a*d^2*x^2+d^2)/d^2)-32/27*(d*x)^(3/2)/d-16/9*d/a/(d^
2/a)^(1/4)*arctan((d*x)^(1/2)/(d^2/a)^(1/4))+8/9*d/a/(d^2/a)^(1/4)*ln(((d*x)^(1/2)+(d^2/a)^(1/4))/((d*x)^(1/2)
-(d^2/a)^(1/4)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.70334, size = 440, normalized size = 3.52 \begin{align*} \frac{2}{27} \, \sqrt{d x}{\left (9 \, x{\rm Li}_2\left (a x^{2}\right ) + 12 \, x \log \left (-a x^{2} + 1\right ) - 16 \, x\right )} + \frac{32}{9} \, \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a d \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} - \sqrt{d^{3} x + a d^{2} \sqrt{\frac{d^{2}}{a^{3}}}} a \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}}}{d^{2}}\right ) + \frac{8}{9} \, \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} \log \left (512 \, a^{2} \left (\frac{d^{2}}{a^{3}}\right )^{\frac{3}{4}} + 512 \, \sqrt{d x} d\right ) - \frac{8}{9} \, \left (\frac{d^{2}}{a^{3}}\right )^{\frac{1}{4}} \log \left (-512 \, a^{2} \left (\frac{d^{2}}{a^{3}}\right )^{\frac{3}{4}} + 512 \, \sqrt{d x} d\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*sqrt(d*x)*(9*x*dilog(a*x^2) + 12*x*log(-a*x^2 + 1) - 16*x) + 32/9*(d^2/a^3)^(1/4)*arctan(-(sqrt(d*x)*a*d*
(d^2/a^3)^(1/4) - sqrt(d^3*x + a*d^2*sqrt(d^2/a^3))*a*(d^2/a^3)^(1/4))/d^2) + 8/9*(d^2/a^3)^(1/4)*log(512*a^2*
(d^2/a^3)^(3/4) + 512*sqrt(d*x)*d) - 8/9*(d^2/a^3)^(1/4)*log(-512*a^2*(d^2/a^3)^(3/4) + 512*sqrt(d*x)*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d 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((d*x)^(1/2)*polylog(2,a*x^2),x, algorithm="giac")

[Out]

integrate(sqrt(d*x)*dilog(a*x^2), x)