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

Optimal. Leaf size=28 \[ \frac{\text{PolyLog}\left (2,\frac{b c-a d}{b (c+d x)}\right )}{d f} \]

[Out]

PolyLog[2, (b*c - a*d)/(b*(c + d*x))]/(d*f)

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Rubi [A]  time = 0.0224578, antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2447} \[ \frac{\text{PolyLog}\left (2,1-\frac{d (a+b x)}{b (c+d x)}\right )}{d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

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

Mathematica [B]  time = 0.050042, size = 114, normalized size = 4.07 \[ \frac{\log \left (\frac{b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-2 \log \left (\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{b c-a d}{b c+b d x}\right )\right )-2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[(b*c - a*d)/(b*c + b*d*x)]*(2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - 2*Log[(d*(a + b*x))/(b*(c + d*x))] + Lo
g[(b*c - a*d)/(b*c + b*d*x)]) - 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*d*f)

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Maple [A]  time = 0.048, size = 30, normalized size = 1.1 \begin{align*}{\frac{1}{df}{\it dilog} \left ( 1+{\frac{ad-bc}{b \left ( dx+c \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.23762, size = 213, normalized size = 7.61 \begin{align*} -\frac{b{\left (\frac{\log \left (d x + c\right )^{2}}{b f} - \frac{2 \,{\left (\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )\right )}}{b f}\right )}}{2 \, d} - \frac{b{\left (\frac{d \log \left (b x + a\right )}{b} - \frac{d \log \left (d x + c\right )}{b}\right )} \log \left (d f x + c f\right )}{d^{2} f} + \frac{\log \left (d f x + c f\right ) \log \left (\frac{{\left (b x + a\right )} d}{{\left (d x + c\right )} b}\right )}{d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*(log(d*x + c)^2/(b*f) - 2*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c
- a*d)))/(b*f))/d - b*(d*log(b*x + a)/b - d*log(d*x + c)/b)*log(d*f*x + c*f)/(d^2*f) + log(d*f*x + c*f)*log((b
*x + a)*d/((d*x + c)*b))/(d*f)

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Fricas [A]  time = 0.974562, size = 63, normalized size = 2.25 \begin{align*} \frac{{\rm Li}_2\left (-\frac{b d x + a d}{b d x + b c} + 1\right )}{d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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