3.342 \(\int \frac {\log (1-\frac {c (a-b x)}{a+b x})}{(a-b x) (a+b x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {\text {Li}_2\left (\frac {c (a-b x)}{a+b x}\right )}{2 a b} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2517, 2502, 2315} \[ \frac {\text {PolyLog}\left (2,\frac {c (a-b x)}{a+b x}\right )}{2 a b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2315

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

Rule 2502

Int[Log[((e_.)*((c_.) + (d_.)*(x_)))/((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{g = Coeff[Simplify[1/(u*(a
+ b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Dist[(b - d*e)/(h*(b*c - a*d)), Subst[Int[Log[
e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)], x] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]] /; FreeQ[{a, b, c, d, e}
, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]

Rule 2517

Int[Log[(e_.)*((f_.)*((g_) + (v_.)/(w_)))^(r_.)]^(s_.)*(u_.), x_Symbol] :> Int[u*Log[e*((f*ExpandToSum[v + g*w
, x])/ExpandToSum[w, x])^r]^s, x] /; FreeQ[{e, f, g, r, s}, x] && LinearQ[w, x] && (FreeQ[v, x] || LinearQ[v,
x]) && AlgebraicFunctionQ[u, x]

Rubi steps

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

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Mathematica [B]  time = 0.16, size = 252, normalized size = 9.33 \[ \frac {2 \text {Li}_2\left (\frac {(c+1) (a-b x)}{2 a}\right )-2 \text {Li}_2\left (\frac {(c+1) (a+b x)}{2 a c}\right )+\log ^2\left (\frac {2 a c}{(c+1) (a+b x)}\right )+2 \log \left (-\frac {a (-c)+a+b (c+1) x}{2 a c}\right ) \log \left (\frac {2 a c}{(c+1) (a+b x)}\right )-2 \log \left (\frac {a (-c)+a+b (c+1) x}{a+b x}\right ) \log \left (\frac {2 a c}{(c+1) (a+b x)}\right )+2 \log (a-b x) \log \left (\frac {a (-c)+a+b (c+1) x}{2 a}\right )-2 \log (a-b x) \log \left (\frac {a (-c)+a+b (c+1) x}{a+b x}\right )-2 \text {Li}_2\left (\frac {a-b x}{2 a}\right )-2 \log (a-b x) \log \left (\frac {a+b x}{2 a}\right )}{4 a b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[(2*a*c)/((1 + c)*(a + b*x))]^2 - 2*Log[a - b*x]*Log[(a + b*x)/(2*a)] + 2*Log[a - b*x]*Log[(a - a*c + b*(1
+ c)*x)/(2*a)] + 2*Log[(2*a*c)/((1 + c)*(a + b*x))]*Log[-1/2*(a - a*c + b*(1 + c)*x)/(a*c)] - 2*Log[a - b*x]*
Log[(a - a*c + b*(1 + c)*x)/(a + b*x)] - 2*Log[(2*a*c)/((1 + c)*(a + b*x))]*Log[(a - a*c + b*(1 + c)*x)/(a + b
*x)] - 2*PolyLog[2, (a - b*x)/(2*a)] + 2*PolyLog[2, ((1 + c)*(a - b*x))/(2*a)] - 2*PolyLog[2, ((1 + c)*(a + b*
x))/(2*a*c)])/(4*a*b)

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fricas [A]  time = 0.42, size = 34, normalized size = 1.26 \[ \frac {{\rm Li}_2\left (\frac {a c - {\left (b c + b\right )} x - a}{b x + a} + 1\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1-c*(-b*x+a)/(b*x+a))/(-b*x+a)/(b*x+a),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1-c*(-b*x+a)/(b*x+a))/(-b*x+a)/(b*x+a),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 24, normalized size = 0.89 \[ \frac {\dilog \left (-\frac {2 a c}{b x +a}+c +1\right )}{2 a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.52, size = 243, normalized size = 9.00 \[ \frac {1}{2} \, {\left (\frac {\log \left (b x + a\right )}{a b} - \frac {\log \left (b x - a\right )}{a b}\right )} \log \left (\frac {{\left (b x - a\right )} c}{b x + a} + 1\right ) + \frac {\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (b x - a\right )}{4 \, a b} + \frac {\log \left (b x - a\right ) \log \left (\frac {b {\left (c + 1\right )} x - a {\left (c + 1\right )}}{2 \, a} + 1\right ) + {\rm Li}_2\left (-\frac {b {\left (c + 1\right )} x - a {\left (c + 1\right )}}{2 \, a}\right )}{2 \, a b} + \frac {\log \left (b x + a\right ) \log \left (-\frac {b x + a}{2 \, a} + 1\right ) + {\rm Li}_2\left (\frac {b x + a}{2 \, a}\right )}{2 \, a b} - \frac {\log \left (b x + a\right ) \log \left (-\frac {b {\left (c + 1\right )} x + a {\left (c + 1\right )}}{2 \, a c} + 1\right ) + {\rm Li}_2\left (\frac {b {\left (c + 1\right )} x + a {\left (c + 1\right )}}{2 \, a c}\right )}{2 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1-c*(-b*x+a)/(b*x+a))/(-b*x+a)/(b*x+a),x, algorithm="maxima")

[Out]

1/2*(log(b*x + a)/(a*b) - log(b*x - a)/(a*b))*log((b*x - a)*c/(b*x + a) + 1) + 1/4*(log(b*x + a)^2 - 2*log(b*x
+ a)*log(b*x - a))/(a*b) + 1/2*(log(b*x - a)*log(1/2*(b*(c + 1)*x - a*(c + 1))/a + 1) + dilog(-1/2*(b*(c + 1)
*x - a*(c + 1))/a))/(a*b) + 1/2*(log(b*x + a)*log(-1/2*(b*x + a)/a + 1) + dilog(1/2*(b*x + a)/a))/(a*b) - 1/2*
(log(b*x + a)*log(-1/2*(b*(c + 1)*x + a*(c + 1))/(a*c) + 1) + dilog(1/2*(b*(c + 1)*x + a*(c + 1))/(a*c)))/(a*b
)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (1-\frac {c\,\left (a-b\,x\right )}{a+b\,x}\right )}{\left (a+b\,x\right )\,\left (a-b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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