3.378 \(\int \frac{\log (a+b x) \log (c+d x)}{x} \, dx\)

Optimal. Leaf size=364 \[ \text{PolyLog}\left (3,\frac{c (a+b x)}{a (c+d x)}\right )-\text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{c (a+b x)}{a (c+d x)}\right )-\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )+\text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{d x}{c}+1\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )-\text{PolyLog}\left (3,\frac{b x}{a}+1\right )-\text{PolyLog}\left (3,\frac{d x}{c}+1\right )+\frac{1}{2} \left (\log \left (\frac{b c-a d}{b (c+d x)}\right )-\log \left (-\frac{x (b c-a d)}{a (c+d x)}\right )+\log \left (-\frac{b x}{a}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )-\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )^2+\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x) \]

[Out]

Log[-((b*x)/a)]*Log[a + b*x]*Log[c + d*x] + ((Log[-((b*x)/a)] + Log[(b*c - a*d)/(b*(c + d*x))] - Log[-(((b*c -
a*d)*x)/(a*(c + d*x)))])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2 - ((Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a
+ b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])^2)/2 + (Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2,
1 + (b*x)/a] + Log[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))] - Log[(a*(c + d*x))/(
c*(a + b*x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] + (Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*Poly
Log[2, 1 + (d*x)/c] - PolyLog[3, 1 + (b*x)/a] + PolyLog[3, (c*(a + b*x))/(a*(c + d*x))] - PolyLog[3, (d*(a + b
*x))/(b*(c + d*x))] - PolyLog[3, 1 + (d*x)/c]

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Rubi [A]  time = 0.0545108, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2435} \[ \text{PolyLog}\left (3,\frac{c (a+b x)}{a (c+d x)}\right )-\text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{c (a+b x)}{a (c+d x)}\right )-\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )+\text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{d x}{c}+1\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )-\text{PolyLog}\left (3,\frac{b x}{a}+1\right )-\text{PolyLog}\left (3,\frac{d x}{c}+1\right )+\frac{1}{2} \left (\log \left (\frac{b c-a d}{b (c+d x)}\right )-\log \left (-\frac{x (b c-a d)}{a (c+d x)}\right )+\log \left (-\frac{b x}{a}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )-\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )^2+\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x) \]

Antiderivative was successfully verified.

[In]

Int[(Log[a + b*x]*Log[c + d*x])/x,x]

[Out]

Log[-((b*x)/a)]*Log[a + b*x]*Log[c + d*x] + ((Log[-((b*x)/a)] + Log[(b*c - a*d)/(b*(c + d*x))] - Log[-(((b*c -
a*d)*x)/(a*(c + d*x)))])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2 - ((Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a
+ b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])^2)/2 + (Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2,
1 + (b*x)/a] + Log[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))] - Log[(a*(c + d*x))/(
c*(a + b*x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] + (Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*Poly
Log[2, 1 + (d*x)/c] - PolyLog[3, 1 + (b*x)/a] + PolyLog[3, (c*(a + b*x))/(a*(c + d*x))] - PolyLog[3, (d*(a + b
*x))/(b*(c + d*x))] - PolyLog[3, 1 + (d*x)/c]

Rule 2435

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x)
)])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1
+ (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Lo
g[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*
x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x
)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int \frac{\log (a+b x) \log (c+d x)}{x} \, dx &=\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )+\log \left (\frac{b c-a d}{b (c+d x)}\right )-\log \left (-\frac{(b c-a d) x}{a (c+d x)}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )-\frac{1}{2} \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log (a+b x)+\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )^2+\left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right ) \text{Li}_2\left (1+\frac{b x}{a}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{Li}_2\left (\frac{c (a+b x)}{a (c+d x)}\right )-\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )+\left (\log (a+b x)+\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right ) \text{Li}_2\left (1+\frac{d x}{c}\right )-\text{Li}_3\left (1+\frac{b x}{a}\right )+\text{Li}_3\left (\frac{c (a+b x)}{a (c+d x)}\right )-\text{Li}_3\left (\frac{d (a+b x)}{b (c+d x)}\right )-\text{Li}_3\left (1+\frac{d x}{c}\right )\\ \end{align*}

Mathematica [A]  time = 0.0943344, size = 394, normalized size = 1.08 \[ \text{PolyLog}\left (3,\frac{a (c+d x)}{c (a+b x)}\right )-\text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )+\log \left (\frac{a (c+d x)}{c (a+b x)}\right ) \left (\text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )-\text{PolyLog}\left (2,\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac{a (c+d x)}{c (a+b x)}\right )\right )+\text{PolyLog}\left (2,\frac{d x}{c}+1\right ) \left (\log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )-\text{PolyLog}\left (3,\frac{b x}{a}+1\right )-\text{PolyLog}\left (3,\frac{d x}{c}+1\right )+\frac{1}{2} \left (\log \left (\frac{a d-b c}{d (a+b x)}\right )-\log \left (\frac{b c x-a d x}{a c+b c x}\right )+\log \left (-\frac{b x}{a}\right )\right ) \log ^2\left (\frac{a (c+d x)}{c (a+b x)}\right )+\log \left (\frac{d x}{c}+1\right ) \left (\log \left (-\frac{d x}{c}\right )-\log \left (-\frac{b x}{a}\right )\right ) \log \left (\frac{a (c+d x)}{c (a+b x)}\right )+\log \left (-\frac{b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac{1}{2} \log \left (\frac{d x}{c}+1\right ) \left (\log \left (-\frac{b x}{a}\right )-\log \left (-\frac{d x}{c}\right )\right ) \left (\log \left (\frac{d x}{c}+1\right )-2 \log (a+b x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[a + b*x]*Log[c + d*x])/x,x]

[Out]

Log[-((b*x)/a)]*Log[a + b*x]*Log[c + d*x] + (Log[(a*(c + d*x))/(c*(a + b*x))]^2*(Log[-((b*x)/a)] + Log[(-(b*c)
+ a*d)/(d*(a + b*x))] - Log[(b*c*x - a*d*x)/(a*c + b*c*x)]))/2 + (-Log[-((b*x)/a)] + Log[-((d*x)/c)])*Log[(a*
(c + d*x))/(c*(a + b*x))]*Log[1 + (d*x)/c] + ((Log[-((b*x)/a)] - Log[-((d*x)/c)])*Log[1 + (d*x)/c]*(-2*Log[a +
b*x] + Log[1 + (d*x)/c]))/2 + (Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (b*x)/a] + Log
[(a*(c + d*x))/(c*(a + b*x))]*(-PolyLog[2, (a*(c + d*x))/(c*(a + b*x))] + PolyLog[2, (b*(c + d*x))/(d*(a + b*x
))]) + (Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c] - PolyLog[3, 1 + (b*x)/a] + P
olyLog[3, (a*(c + d*x))/(c*(a + b*x))] - PolyLog[3, (b*(c + d*x))/(d*(a + b*x))] - PolyLog[3, 1 + (d*x)/c]

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( bx+a \right ) \ln \left ( dx+c \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(b*x+a)*ln(d*x+c)/x,x)

[Out]

int(ln(b*x+a)*ln(d*x+c)/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (b x + a\right ) \log \left (d x + c\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(b*x+a)*log(d*x+c)/x,x, algorithm="maxima")

[Out]

integrate(log(b*x + a)*log(d*x + c)/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (b x + a\right ) \log \left (d x + c\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(b*x+a)*log(d*x+c)/x,x, algorithm="fricas")

[Out]

integral(log(b*x + a)*log(d*x + c)/x, x)

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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(ln(b*x+a)*ln(d*x+c)/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (b x + a\right ) \log \left (d x + c\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(b*x+a)*log(d*x+c)/x,x, algorithm="giac")

[Out]

integrate(log(b*x + a)*log(d*x + c)/x, x)