3.2.6 \(\int \frac {\log (\frac {d (a+b x)}{b (c+d x)})}{c f+d f x} \, dx\) [106]

Optimal. Leaf size=28 \[ \frac {\text {Li}_2\left (\frac {b c-a d}{b (c+d x)}\right )}{d f} \]

[Out]

polylog(2,(-a*d+b*c)/b/(d*x+c))/d/f

________________________________________________________________________________________

Rubi [A]
time = 0.02, 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 = {2497} \begin {gather*} \frac {\text {PolyLog}\left (2,1-\frac {d (a+b x)}{b (c+d x)}\right )}{d f} \end {gather*}

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 2497

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] Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(28)=56\).
time = 0.04, size = 114, normalized size = 4.07 \begin {gather*} \frac {\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\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 {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 d f} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.23, size = 30, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\dilog \left (1+\frac {a d -c b}{b \left (d x +c \right )}\right )}{d f}\)	\(30\)
default	\(\frac {\dilog \left (1+\frac {a d -c b}{b \left (d x +c \right )}\right )}{d f}\)	\(30\)
risch	\(\frac {\dilog \left (1+\frac {a d -c b}{b \left (d x +c \right )}\right )}{d f}\)	\(30\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (27) = 54\).
time = 0.28, size = 158, normalized size = 5.64 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 30, normalized size = 1.07 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {b d x + a d}{b d x + b c} + 1\right )}{d f} \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\log {\left (\frac {a d}{b c + b d x} + \frac {b d x}{b c + b d x} \right )}}{c + d x}\, dx}{f} \end {gather*}

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]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1203 vs. \(2 (27) = 54\).
time = 35.39, size = 1203, normalized size = 42.96 \begin {gather*} -\frac {1}{2} \, {\left (\frac {b^{2} c d}{{\left (b c - a d\right )}^{2}} - \frac {a b d^{2}}{{\left (b c - a d\right )}^{2}}\right )}^{2} {\left ({\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\frac {\log \left (\frac {{\left | b d x + a d \right |}}{{\left | b d x + b c \right |}}\right )}{b^{3} d^{4} f} - \frac {\log \left ({\left | \frac {b d x + a d}{b d x + b c} - 1 \right |}\right )}{b^{3} d^{4} f} - \frac {1}{b^{3} d^{4} f {\left (\frac {b d x + a d}{b d x + b c} - 1\right )}}\right )} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\frac {{\left (a + \frac {b {\left (\frac {{\left (a d - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )} b c}{{\left (b c - a d\right )} {\left (b c - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )}} - \frac {a d}{b c - a d}\right )}}{\frac {b d}{b c - a d} - \frac {{\left (a d - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )} b d}{{\left (b c - a d\right )} {\left (b c - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )}}}\right )} d}{b {\left (c + \frac {{\left (\frac {{\left (a d - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )} b c}{{\left (b c - a d\right )} {\left (b c - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {b d}{b c - a d} - \frac {{\left (a d - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )} b d}{{\left (b c - a d\right )} {\left (b c - \frac {b {\left (\frac {{\left (b d x + a d\right )} b c}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {a d}{b c - a d}\right )} d}{\frac {{\left (b d x + a d\right )} b d}{{\left (b d x + b c\right )} {\left (b c - a d\right )}} - \frac {b d}{b c - a d}}\right )}}}\right )}}\right )}{b^{3} d^{4} f {\left (\frac {b d x + a d}{b d x + b c} - 1\right )}^{2}}\right )} \end {gather*}

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]

-1/2*(b^2*c*d/(b*c - a*d)^2 - a*b*d^2/(b*c - a*d)^2)^2*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(l
og(abs(b*d*x + a*d)/abs(b*d*x + b*c))/(b^3*d^4*f) - log(abs((b*d*x + a*d)/(b*d*x + b*c) - 1))/(b^3*d^4*f) - 1/
(b^3*d^4*f*((b*d*x + a*d)/(b*d*x + b*c) - 1))) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log((a +
b*((a*d - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b
*c)*(b*c - a*d)) - b*d/(b*c - a*d)))*b*c/((b*c - a*d)*(b*c - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d))
- a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))) - a*d/(b*c - a*d))/(b
*d/(b*c - a*d) - (a*d - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b
*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))*b*d/((b*c - a*d)*(b*c - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)
*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d))))))*d/(b*
(c + ((a*d - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x
+ b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))*b*c/((b*c - a*d)*(b*c - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d
)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))) - a*d/(b*c - a*d))
*d/(b*d/(b*c - a*d) - (a*d - b*((b*d*x + a*d)*b*c/((b*d*x + b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a
*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))*b*d/((b*c - a*d)*(b*c - b*((b*d*x + a*d)*b*c/((b*d*x +
b*c)*(b*c - a*d)) - a*d/(b*c - a*d))*d/((b*d*x + a*d)*b*d/((b*d*x + b*c)*(b*c - a*d)) - b*d/(b*c - a*d)))))))
)/(b^3*d^4*f*((b*d*x + a*d)/(b*d*x + b*c) - 1)^2))

________________________________________________________________________________________

Mupad [B]
time = 4.25, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {d\,\left (a+b\,x\right )}{b\,\left (c+d\,x\right )}\right )}{d\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________