3.4.78 \(\int \frac {\log (a+b x) \log (c+d x)}{x} \, dx\) [378]

Optimal. Leaf size=364 \[ \log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {b c-a d}{b (c+d x)}\right )-\log \left (-\frac {(b c-a d) x}{a (c+d x)}\right )\right ) \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right )-\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \left (\log (a+b x)+\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right )^2+\left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )+\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \text {Li}_2\left (\frac {c (a+b x)}{a (c+d x)}\right )-\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )+\left (\log (a+b x)+\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \text {Li}_2\left (1+\frac {d x}{c}\right )-\text {Li}_3\left (1+\frac {b x}{a}\right )+\text {Li}_3\left (\frac {c (a+b x)}{a (c+d x)}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )-\text {Li}_3\left (1+\frac {d x}{c}\right ) \]

[Out]

ln(-b*x/a)*ln(b*x+a)*ln(d*x+c)+1/2*(ln(-b*x/a)+ln((-a*d+b*c)/b/(d*x+c))-ln(-(-a*d+b*c)*x/a/(d*x+c)))*ln(a*(d*x
+c)/c/(b*x+a))^2-1/2*(ln(-b*x/a)-ln(-d*x/c))*(ln(b*x+a)+ln(a*(d*x+c)/c/(b*x+a)))^2+(ln(d*x+c)-ln(a*(d*x+c)/c/(
b*x+a)))*polylog(2,1+b*x/a)+ln(a*(d*x+c)/c/(b*x+a))*polylog(2,c*(b*x+a)/a/(d*x+c))-ln(a*(d*x+c)/c/(b*x+a))*pol
ylog(2,d*(b*x+a)/b/(d*x+c))+(ln(b*x+a)+ln(a*(d*x+c)/c/(b*x+a)))*polylog(2,1+d*x/c)-polylog(3,1+b*x/a)+polylog(
3,c*(b*x+a)/a/(d*x+c))-polylog(3,d*(b*x+a)/b/(d*x+c))-polylog(3,1+d*x/c)

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Rubi [A]
time = 0.04, antiderivative size = 364, normalized size of antiderivative = 1.00, 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 = {2485} \begin {gather*} \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) \end {gather*}

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 2485

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[(1/2)*(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, x] - Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-d)*(x/c)])*(Log[a + b*x] +
Log[a*((c + d*x)/(c*(a + b*x)))])^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*}

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Mathematica [A]
time = 0.04, size = 394, normalized size = 1.08 \begin {gather*} \log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac {1}{2} \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right ) \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {-b c+a d}{d (a+b x)}\right )-\log \left (\frac {b c x-a d x}{a c+b c x}\right )\right )+\left (-\log \left (-\frac {b x}{a}\right )+\log \left (-\frac {d x}{c}\right )\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \log \left (1+\frac {d x}{c}\right )+\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \log \left (1+\frac {d x}{c}\right ) \left (-2 \log (a+b x)+\log \left (1+\frac {d x}{c}\right )\right )+\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 ) \left (-\text {Li}_2\left (\frac {a (c+d x)}{c (a+b x)}\right )+\text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )\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 {a (c+d x)}{c (a+b x)}\right )-\text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )-\text {Li}_3\left (1+\frac {d x}{c}\right ) \end {gather*}

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 = 0.39, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (b x +a \right ) \ln \left (d x +c \right )}{x}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, 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(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, 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(log(b*x+a)*log(d*x+c)/x,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(a + b*x)*log(c + d*x))/x,x)

[Out]

int((log(a + b*x)*log(c + d*x))/x, x)

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