Integrand size = 34, antiderivative size = 27 \[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {c (a-b x)}{a+b x}\right )}{2 a b} \] Output:
1/2*polylog(2,c*(-b*x+a)/(b*x+a))/a/b
Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(27)=54\).
Time = 0.35 (sec) , antiderivative size = 259, normalized size of antiderivative = 9.59 \[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {4 \text {arctanh}\left (\frac {b x}{a}\right ) \log \left (\frac {a}{b}+x\right )-\log ^2\left (\frac {a}{b}+x\right )-4 \text {arctanh}\left (\frac {b x}{a}\right ) \log \left (\frac {a-a c}{b+b c}+x\right )+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {a-b x}{2 a}\right )-2 \log \left (\frac {a-a c}{b+b c}+x\right ) \log \left (\frac {(1+c) (a-b x)}{2 a}\right )+2 \log \left (\frac {a-a c}{b+b c}+x\right ) \log \left (\frac {(1+c) (a+b x)}{2 a c}\right )+4 \text {arctanh}\left (\frac {b x}{a}\right ) \log \left (\frac {a-a c+b (1+c) x}{a+b x}\right )+2 \operatorname {PolyLog}\left (2,\frac {a+b x}{2 a}\right )-2 \operatorname {PolyLog}\left (2,\frac {a-a c+b (1+c) x}{2 a}\right )+2 \operatorname {PolyLog}\left (2,-\frac {a-a c+b (1+c) x}{2 a c}\right )}{4 a b} \] Input:
Integrate[Log[1 - (c*(a - b*x))/(a + b*x)]/(a^2 - b^2*x^2),x]
Output:
(4*ArcTanh[(b*x)/a]*Log[a/b + x] - Log[a/b + x]^2 - 4*ArcTanh[(b*x)/a]*Log [(a - a*c)/(b + b*c) + x] + 2*Log[a/b + x]*Log[(a - b*x)/(2*a)] - 2*Log[(a - a*c)/(b + b*c) + x]*Log[((1 + c)*(a - b*x))/(2*a)] + 2*Log[(a - a*c)/(b + b*c) + x]*Log[((1 + c)*(a + b*x))/(2*a*c)] + 4*ArcTanh[(b*x)/a]*Log[(a - a*c + b*(1 + c)*x)/(a + b*x)] + 2*PolyLog[2, (a + b*x)/(2*a)] - 2*PolyLo g[2, (a - a*c + b*(1 + c)*x)/(2*a)] + 2*PolyLog[2, -1/2*(a - a*c + b*(1 + c)*x)/(a*c)])/(4*a*b)
Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2897}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle \frac {\operatorname {PolyLog}\left (2,\frac {c (a-b x)}{a+b x}\right )}{2 a b}\) |
Input:
Int[Log[1 - (c*(a - b*x))/(a + b*x)]/(a^2 - b^2*x^2),x]
Output:
PolyLog[2, (c*(a - b*x))/(a + b*x)]/(2*a*b)
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[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Time = 2.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\operatorname {dilog}\left (1+c -\frac {2 a c}{b x +a}\right )}{2 b a}\) | \(24\) |
default | \(\frac {\operatorname {dilog}\left (1+c -\frac {2 a c}{b x +a}\right )}{2 b a}\) | \(24\) |
risch | \(\frac {\operatorname {dilog}\left (1+c -\frac {2 a c}{b x +a}\right )}{2 b a}\) | \(24\) |
parts | \(-\frac {\ln \left (1-\frac {c \left (-b x +a \right )}{b x +a}\right ) \ln \left (-b x +a \right )}{2 a b}+\frac {\ln \left (1-\frac {c \left (-b x +a \right )}{b x +a}\right ) \ln \left (b x +a \right )}{2 a b}+\frac {c \left (-\frac {\operatorname {dilog}\left (-\frac {-b x -a}{2 a}\right )+\ln \left (-b x +a \right ) \ln \left (-\frac {-b x -a}{2 a}\right )}{2 a c}+\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (1+c \right ) \left (-b x +a \right )-2 a}{2 a}\right )}{1+c}+\frac {\ln \left (-b x +a \right ) \ln \left (-\frac {\left (1+c \right ) \left (-b x +a \right )-2 a}{2 a}\right )}{1+c}\right ) \left (1+c \right )}{2 a c}\right )}{b}+\frac {c \left (\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (1+c \right ) \left (b x +a \right )-2 a c}{2 a c}\right )}{1+c}+\frac {\ln \left (b x +a \right ) \ln \left (-\frac {\left (1+c \right ) \left (b x +a \right )-2 a c}{2 a c}\right )}{1+c}\right ) \left (-1-c \right )}{2 a c}+\frac {\ln \left (b x +a \right )^{2}}{4 a c}\right )}{b}\) | \(291\) |
Input:
int(ln(1-c*(-b*x+a)/(b*x+a))/(-b^2*x^2+a^2),x,method=_RETURNVERBOSE)
Output:
1/2/b/a*dilog(1+c-2*a*c/(b*x+a))
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {{\rm Li}_2\left (\frac {a c - {\left (b c + b\right )} x - a}{b x + a} + 1\right )}{2 \, a b} \] Input:
integrate(log(1-c*(-b*x+a)/(b*x+a))/(-b^2*x^2+a^2),x, algorithm="fricas")
Output:
1/2*dilog((a*c - (b*c + b)*x - a)/(b*x + a) + 1)/(a*b)
Timed out. \[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\text {Timed out} \] Input:
integrate(ln(1-c*(-b*x+a)/(b*x+a))/(-b**2*x**2+a**2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (26) = 52\).
Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 9.00 \[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {1}{2} \, {\left (\frac {\log \left (b x + a\right )}{a b} - \frac {\log \left (b x - a\right )}{a b}\right )} \log \left (\frac {{\left (b x - a\right )} c}{b x + a} + 1\right ) + \frac {\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (b x - a\right )}{4 \, a b} + \frac {\log \left (b x - a\right ) \log \left (\frac {b {\left (c + 1\right )} x - a {\left (c + 1\right )}}{2 \, a} + 1\right ) + {\rm Li}_2\left (-\frac {b {\left (c + 1\right )} x - a {\left (c + 1\right )}}{2 \, a}\right )}{2 \, a b} + \frac {\log \left (b x + a\right ) \log \left (-\frac {b x + a}{2 \, a} + 1\right ) + {\rm Li}_2\left (\frac {b x + a}{2 \, a}\right )}{2 \, a b} - \frac {\log \left (b x + a\right ) \log \left (-\frac {b {\left (c + 1\right )} x + a {\left (c + 1\right )}}{2 \, a c} + 1\right ) + {\rm Li}_2\left (\frac {b {\left (c + 1\right )} x + a {\left (c + 1\right )}}{2 \, a c}\right )}{2 \, a b} \] Input:
integrate(log(1-c*(-b*x+a)/(b*x+a))/(-b^2*x^2+a^2),x, algorithm="maxima")
Output:
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) + di log(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)
\[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\int { -\frac {\log \left (\frac {{\left (b x - a\right )} c}{b x + a} + 1\right )}{b^{2} x^{2} - a^{2}} \,d x } \] Input:
integrate(log(1-c*(-b*x+a)/(b*x+a))/(-b^2*x^2+a^2),x, algorithm="giac")
Output:
integrate(-log((b*x - a)*c/(b*x + a) + 1)/(b^2*x^2 - a^2), x)
Timed out. \[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\int \frac {\ln \left (1-\frac {c\,\left (a-b\,x\right )}{a+b\,x}\right )}{a^2-b^2\,x^2} \,d x \] Input:
int(log(1 - (c*(a - b*x))/(a + b*x))/(a^2 - b^2*x^2),x)
Output:
int(log(1 - (c*(a - b*x))/(a + b*x))/(a^2 - b^2*x^2), x)
\[ \int \frac {\log \left (1-\frac {c (a-b x)}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {-8 \left (\int \frac {\mathrm {log}\left (\frac {b c x -a c +b x +a}{b x +a}\right )}{b^{3} c \,x^{3}-a \,b^{2} c \,x^{2}+b^{3} x^{3}-a^{2} b c x +a \,b^{2} x^{2}+a^{3} c -a^{2} b x -a^{3}}d x \right ) a^{2} b c -\mathrm {log}\left (\frac {b c x -a c +b x +a}{b x +a}\right )^{2} c -\mathrm {log}\left (\frac {b c x -a c +b x +a}{b x +a}\right )^{2}}{4 a b c} \] Input:
int(log(1-c*(-b*x+a)/(b*x+a))/(-b^2*x^2+a^2),x)
Output:
( - 8*int(log(( - a*c + a + b*c*x + b*x)/(a + b*x))/(a**3*c - a**3 - a**2* b*c*x - a**2*b*x - a*b**2*c*x**2 + a*b**2*x**2 + b**3*c*x**3 + b**3*x**3), x)*a**2*b*c - log(( - a*c + a + b*c*x + b*x)/(a + b*x))**2*c - log(( - a*c + a + b*c*x + b*x)/(a + b*x))**2)/(4*a*b*c)