3.2.28 \(\int \frac {\text {PolyLog}(2,c (a+b x))}{x^2} \, dx\) [128]

Optimal. Leaf size=84 \[ -\frac {b \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac {b \text {PolyLog}(2,c (a+b x))}{a}-\frac {\text {PolyLog}(2,c (a+b x))}{x}-\frac {b \text {PolyLog}\left (2,1-\frac {b c x}{1-a c}\right )}{a} \]

[Out]

-b*ln(b*c*x/(-a*c+1))*ln(-b*c*x-a*c+1)/a-b*polylog(2,c*(b*x+a))/a-polylog(2,c*(b*x+a))/x-b*polylog(2,1-b*c*x/(
-a*c+1))/a

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Rubi [A]
time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {6733, 36, 29, 31, 2463, 2441, 2352, 2440, 2438} \begin {gather*} -\frac {b \text {Li}_2(c (a+b x))}{a}-\frac {\text {Li}_2(c (a+b x))}{x}-\frac {b \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b \log \left (\frac {b c x}{1-a c}\right ) \log (-a c-b c x+1)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, c*(a + b*x)]/x^2,x]

[Out]

-((b*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/a) - (b*PolyLog[2, c*(a + b*x)])/a - PolyLog[2, c*(a + b*x)]
/x - (b*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 6733

Int[((d_.) + (e_.)*(x_))^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> Simp[(d + e*x)^(m + 1)*(Po
lyLog[2, c*(a + b*x)]/(e*(m + 1))), x] + Dist[b/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(Log[1 - a*c - b*c*x]/(a +
b*x)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 73, normalized size = 0.87 \begin {gather*} -\frac {(a+b x) \text {PolyLog}(2,c (a+b x))+b x \left (\log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)+\text {PolyLog}\left (2,\frac {-1+a c+b c x}{-1+a c}\right )\right )}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, c*(a + b*x)]/x^2,x]

[Out]

-(((a + b*x)*PolyLog[2, c*(a + b*x)] + b*x*(Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x] + PolyLog[2, (-1 + a*c
+ b*c*x)/(-1 + a*c)]))/(a*x))

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Maple [A]
time = 1.31, size = 94, normalized size = 1.12

method	result	size
derivativedivides	\(c b \left (-\frac {\polylog \left (2, x b c +a c \right )}{x b c}-\frac {\dilog \left (-\frac {x b c}{a c -1}\right )+\ln \left (-x b c -a c +1\right ) \ln \left (-\frac {x b c}{a c -1}\right )}{a c}-\frac {\dilog \left (-x b c -a c +1\right )}{a c}\right )\)	\(94\)
default	\(c b \left (-\frac {\polylog \left (2, x b c +a c \right )}{x b c}-\frac {\dilog \left (-\frac {x b c}{a c -1}\right )+\ln \left (-x b c -a c +1\right ) \ln \left (-\frac {x b c}{a c -1}\right )}{a c}-\frac {\dilog \left (-x b c -a c +1\right )}{a c}\right )\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,c*(b*x+a))/x^2,x,method=_RETURNVERBOSE)

[Out]

c*b*(-1/x/b/c*polylog(2,b*c*x+a*c)-(dilog(-x*b*c/(a*c-1))+ln(-b*c*x-a*c+1)*ln(-x*b*c/(a*c-1)))/a/c-dilog(-b*c*
x-a*c+1)/a/c)

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Maxima [A]
time = 0.28, size = 114, normalized size = 1.36 \begin {gather*} \frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b}{a} - \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (-\frac {b c x + a c - 1}{a c - 1} + 1\right ) + {\rm Li}_2\left (\frac {b c x + a c - 1}{a c - 1}\right )\right )} b}{a} - \frac {{\rm Li}_2\left (b c x + a c\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/x^2,x, algorithm="maxima")

[Out]

(log(b*c*x + a*c)*log(-b*c*x - a*c + 1) + dilog(-b*c*x - a*c + 1))*b/a - (log(-b*c*x - a*c + 1)*log(-(b*c*x +
a*c - 1)/(a*c - 1) + 1) + dilog((b*c*x + a*c - 1)/(a*c - 1)))*b/a - dilog(b*c*x + a*c)/x

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Fricas [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,c*(b*x+a))/x^2,x, algorithm="fricas")

[Out]

integral(dilog(b*c*x + a*c)/x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{2}\left (a c + b c x\right )}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,c*(b*x+a))/x**2,x)

[Out]

Integral(polylog(2, a*c + b*c*x)/x**2, x)

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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,c*(b*x+a))/x^2,x, algorithm="giac")

[Out]

integrate(dilog((b*x + a)*c)/x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, c*(a + b*x))/x^2,x)

[Out]

int(polylog(2, c*(a + b*x))/x^2, x)

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