\(\int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx\) [60]

Optimal result

Integrand size = 13, antiderivative size = 102 \[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=-\frac {8 \sqrt {d x}}{9 a}-\frac {8 (d x)^{3/2}}{27 d}+\frac {8 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/2}}+\frac {4 (d x)^{3/2} \log (1-a x)}{9 d}+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=\frac {2 \sqrt {d x} \left (\frac {12 \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}+\frac {2 \sqrt {x} (-6-2 a x+3 a x \log (1-a x))}{a}+9 x^{3/2} \operatorname {PolyLog}(2,a x)\right )}{27 \sqrt {x}} \] Input:

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

Output:

(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])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {7145, 25, 2842, 60, 60, 73, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx\)

\(\Big \downarrow \) 7145

\(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}-\frac {2}{3} \int -\sqrt {d x} \log (1-a x)dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{3} \int \sqrt {d x} \log (1-a x)dx+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {2}{3} \left (\frac {2 a \int \frac {(d x)^{3/2}}{1-a x}dx}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2}{3} \left (\frac {2 a \left (\frac {d \int \frac {\sqrt {d x}}{1-a x}dx}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2}{3} \left (\frac {2 a \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {d x} (1-a x)}dx}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2}{3} \left (\frac {2 a \left (\frac {d \left (\frac {2 \int \frac {1}{1-a x}d\sqrt {d x}}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2}{3} \left (\frac {2 a \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{a^{3/2}}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 2842

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] - Simp[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 7145

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo 
l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[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]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} \operatorname {polylog}\left (2, x a \right )}{3}+\frac {4 \left (d x \right )^{\frac {3}{2}} \ln \left (\frac {-d x a +d}{d}\right )}{9}+\frac {8 a \left (-\frac {\frac {\left (d x \right )^{\frac {3}{2}} a}{3}+d \sqrt {d x}}{a^{2}}+\frac {d^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x}\, a}{\sqrt {a d}}\right )}{a^{2} \sqrt {a d}}\right )}{9}}{d}\)	\(88\)
default	\(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} \operatorname {polylog}\left (2, x a \right )}{3}+\frac {4 \left (d x \right )^{\frac {3}{2}} \ln \left (\frac {-d x a +d}{d}\right )}{9}+\frac {8 a \left (-\frac {\frac {\left (d x \right )^{\frac {3}{2}} a}{3}+d \sqrt {d x}}{a^{2}}+\frac {d^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x}\, a}{\sqrt {a d}}\right )}{a^{2} \sqrt {a d}}\right )}{9}}{d}\)	\(88\)
meijerg	\(\frac {\sqrt {d x}\, \left (-\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (20 x a +60\right )}{135 a^{2}}-\frac {4 \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (\ln \left (1-\sqrt {x a}\right )-\ln \left (1+\sqrt {x a}\right )\right )}{9 a^{2} \sqrt {x a}}+\frac {4 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \ln \left (-x a +1\right )}{9 a}+\frac {2 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \operatorname {polylog}\left (2, x a \right )}{3 a}\right )}{\sqrt {x}\, \sqrt {-a}\, a}\)	\(115\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.40 \[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=\left [\frac {2 \, {\left ({\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} + 6 \, \sqrt {\frac {d}{a}} \log \left (\frac {a d x + 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right )\right )}}{27 \, a}, \frac {2 \, {\left ({\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} - 12 \, \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right )\right )}}{27 \, a}\right ] \] Input:

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

Output:

[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*sqrt( 
-d/a)*arctan(sqrt(d*x)*a*sqrt(-d/a)/d))/a]
 

Sympy [F]

\[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=\int \sqrt {d x} \operatorname {Li}_{2}\left (a x\right )\, dx \] Input:

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

Output:

Integral(sqrt(d*x)*polylog(2, a*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07 \[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=-\frac {2 \, {\left (\frac {6 \, d^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a} - \frac {9 \, \left (d x\right )^{\frac {3}{2}} a {\rm Li}_2\left (a x\right ) + 6 \, \left (d x\right )^{\frac {3}{2}} a \log \left (-a d x + d\right ) - 2 \, \left (d x\right )^{\frac {3}{2}} {\left (3 \, a \log \left (d\right ) + 2 \, a\right )} - 12 \, \sqrt {d x} d}{a}\right )}}{27 \, d} \] Input:

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

Output:

-2/27*(6*d^2*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/(sqr 
t(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
 

Giac [F]

\[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=\int { \sqrt {d x} {\rm Li}_2\left (a x\right ) \,d x } \] Input:

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

Output:

integrate(sqrt(d*x)*dilog(a*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=\int \sqrt {d\,x}\,\mathrm {polylog}\left (2,a\,x\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {d x} \operatorname {PolyLog}(2,a x) \, dx=\sqrt {d}\, \left (\int \sqrt {x}\, \mathit {polylog}\left (2, a x \right )d x \right ) \] Input:

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

Output:

sqrt(d)*int(sqrt(x)*polylog(2,a*x),x)