3.60 \(\int \sqrt {d x} \text {Li}_2(a x) \, dx\)

Optimal. Leaf size=102 \[ \frac {8 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/2}}+\frac {2 (d x)^{3/2} \text {Li}_2(a x)}{3 d}-\frac {8 \sqrt {d x}}{9 a}+\frac {4 (d x)^{3/2} \log (1-a x)}{9 d}-\frac {8 (d x)^{3/2}}{27 d} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {2 (d x)^{3/2} \text {PolyLog}(2,a x)}{3 d}+\frac {8 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/2}}-\frac {8 \sqrt {d x}}{9 a}+\frac {4 (d x)^{3/2} \log (1-a x)}{9 d}-\frac {8 (d x)^{3/2}}{27 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*x]*PolyLog[2, a*x],x]

[Out]

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

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])

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

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Mathematica [A]  time = 0.08, size = 75, normalized size = 0.74 \[ \frac {2 \sqrt {d x} \left (\frac {12 \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}+9 x^{3/2} \text {Li}_2(a x)+\frac {2 \sqrt {x} (-2 a x+3 a x \log (1-a x)-6)}{a}\right )}{27 \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 143, normalized size = 1.40 \[ \left [\frac {2 \, {\left ({\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} + 6 \, \sqrt {\frac {d}{a}} \log \left (\frac {a d x + 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right )\right )}}{27 \, a}, \frac {2 \, {\left ({\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} - 12 \, \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right )\right )}}{27 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x)*dilog(a*x), x)

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maple [A]  time = 0.01, size = 83, normalized size = 0.81 \[ \frac {2 \left (d x \right )^{\frac {3}{2}} \polylog \left (2, a x \right )}{3 d}+\frac {4 \left (d x \right )^{\frac {3}{2}} \ln \left (\frac {-a d x +d}{d}\right )}{9 d}-\frac {8 \left (d x \right )^{\frac {3}{2}}}{27 d}-\frac {8 \sqrt {d x}}{9 a}+\frac {8 d \arctanh \left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{9 a \sqrt {a d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.59, size = 109, normalized size = 1.07 \[ -\frac {2 \, {\left (\frac {6 \, d^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a} - \frac {9 \, \left (d x\right )^{\frac {3}{2}} a {\rm Li}_2\left (a x\right ) + 6 \, \left (d x\right )^{\frac {3}{2}} a \log \left (-a d x + d\right ) - 2 \, \left (d x\right )^{\frac {3}{2}} {\left (3 \, a \log \relax (d) + 2 \, a\right )} - 12 \, \sqrt {d x} d}{a}\right )}}{27 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {d\,x}\,\mathrm {polylog}\left (2,a\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(1/2)*polylog(2, a*x),x)

[Out]

int((d*x)^(1/2)*polylog(2, a*x), x)

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sympy [A]  time = 67.85, size = 119, normalized size = 1.17 \[ \frac {2 \left (\begin {cases} - \frac {2 \left (d x\right )^{\frac {3}{2}} \operatorname {Li}_{1}\left (a x\right )}{9} + \frac {\left (d x\right )^{\frac {3}{2}} \operatorname {Li}_{2}\left (a x\right )}{3} - \frac {4 \left (d x\right )^{\frac {3}{2}}}{27} - \frac {4 d \sqrt {d x}}{9 a} - \frac {4 d^{\frac {3}{2}} \log {\left (- \sqrt {d} \sqrt {\frac {1}{a}} + \sqrt {d x} \right )}}{9 a^{2} \sqrt {\frac {1}{a}}} - \frac {2 d^{\frac {3}{2}} \operatorname {Li}_{1}\left (a x\right )}{9 a^{2} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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