3.1.8 \(\int \frac {\text {PolyLog}(2,a x)}{x^3} \, dx\) [8]

Optimal. Leaf size=58 \[ -\frac {a}{4 x}+\frac {1}{4} a^2 \log (x)-\frac {1}{4} a^2 \log (1-a x)+\frac {\log (1-a x)}{4 x^2}-\frac {\text {PolyLog}(2,a x)}{2 x^2} \]

[Out]

-1/4*a/x+1/4*a^2*ln(x)-1/4*a^2*ln(-a*x+1)+1/4*ln(-a*x+1)/x^2-1/2*polylog(2,a*x)/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6726, 2442, 46} \begin {gather*} \frac {1}{4} a^2 \log (x)-\frac {1}{4} a^2 \log (1-a x)-\frac {\text {Li}_2(a x)}{2 x^2}+\frac {\log (1-a x)}{4 x^2}-\frac {a}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/x^3,x]

[Out]

-1/4*a/x + (a^2*Log[x])/4 - (a^2*Log[1 - a*x])/4 + Log[1 - a*x]/(4*x^2) - PolyLog[2, a*x]/(2*x^2)

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2442

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 6726

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 50, normalized size = 0.86 \begin {gather*} \frac {-a x+a^2 x^2 \log (x)+\log (1-a x)-a^2 x^2 \log (1-a x)-2 \text {PolyLog}(2,a x)}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x]/x^3,x]

[Out]

(-(a*x) + a^2*x^2*Log[x] + Log[1 - a*x] - a^2*x^2*Log[1 - a*x] - 2*PolyLog[2, a*x])/(4*x^2)

________________________________________________________________________________________

Maple [A]
time = 0.44, size = 61, normalized size = 1.05

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\polylog \left (2, a x \right )}{2 a^{2} x^{2}}-\frac {1}{4 a x}+\frac {\ln \left (-a x \right )}{4}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right ) \left (-a x -1\right )}{4 a^{2} x^{2}}\right )\)	\(61\)
default	\(a^{2} \left (-\frac {\polylog \left (2, a x \right )}{2 a^{2} x^{2}}-\frac {1}{4 a x}+\frac {\ln \left (-a x \right )}{4}-\frac {\ln \left (-a x +1\right ) \left (-a x +1\right ) \left (-a x -1\right )}{4 a^{2} x^{2}}\right )\)	\(61\)
meijerg	\(-a^{2} \left (-\frac {9 a x +27}{36 a x}-\frac {\left (-9 a^{2} x^{2}+9\right ) \ln \left (-a x +1\right )}{36 a^{2} x^{2}}+\frac {\polylog \left (2, a x \right )}{2 a^{2} x^{2}}+\frac {1}{4}-\frac {\ln \left (x \right )}{4}-\frac {\ln \left (-a \right )}{4}+\frac {1}{a x}\right )\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/2/a^2/x^2*polylog(2,a*x)-1/4/a/x+1/4*ln(-a*x)-1/4*ln(-a*x+1)*(-a*x+1)*(-a*x-1)/a^2/x^2)

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 40, normalized size = 0.69 \begin {gather*} \frac {1}{4} \, a^{2} \log \left (x\right ) - \frac {a x + {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right ) + 2 \, {\rm Li}_2\left (a x\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x^3,x, algorithm="maxima")

[Out]

1/4*a^2*log(x) - 1/4*(a*x + (a^2*x^2 - 1)*log(-a*x + 1) + 2*dilog(a*x))/x^2

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 47, normalized size = 0.81 \begin {gather*} -\frac {a^{2} x^{2} \log \left (a x - 1\right ) - a^{2} x^{2} \log \left (x\right ) + a x + 2 \, {\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right )}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x^3,x, algorithm="fricas")

[Out]

-1/4*(a^2*x^2*log(a*x - 1) - a^2*x^2*log(x) + a*x + 2*dilog(a*x) - log(-a*x + 1))/x^2

________________________________________________________________________________________

Sympy [A]
time = 0.67, size = 42, normalized size = 0.72 \begin {gather*} \frac {a^{2} \log {\left (x \right )}}{4} + \frac {a^{2} \operatorname {Li}_{1}\left (a x\right )}{4} - \frac {a}{4 x} - \frac {\operatorname {Li}_{1}\left (a x\right )}{4 x^{2}} - \frac {\operatorname {Li}_{2}\left (a x\right )}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x**3,x)

[Out]

a**2*log(x)/4 + a**2*polylog(1, a*x)/4 - a/(4*x) - polylog(1, a*x)/(4*x**2) - polylog(2, a*x)/(2*x**2)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x^3,x, algorithm="giac")

[Out]

integrate(dilog(a*x)/x^3, x)

________________________________________________________________________________________

Mupad [B]
time = 0.30, size = 51, normalized size = 0.88 \begin {gather*} \frac {a^2\,\ln \left (x\right )}{2}+\frac {\ln \left (1-a\,x\right )}{4\,x^2}-\frac {a^2\,\ln \left (a\,x^2-x\right )}{4}-\frac {a}{4\,x}-\frac {\mathrm {polylog}\left (2,a\,x\right )}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x)/x^3,x)

[Out]

(a^2*log(x))/2 + log(1 - a*x)/(4*x^2) - (a^2*log(a*x^2 - x))/4 - a/(4*x) - polylog(2, a*x)/(2*x^2)

________________________________________________________________________________________