Integrand size = 38, antiderivative size = 37 \[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {\operatorname {PolyLog}\left (2,1-\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{2 a b} \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2497} \[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {\operatorname {PolyLog}\left (2,1-\frac {a (1-c)+b (c+1) x}{a+b x}\right )}{2 a b} \]
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Rule 2497
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Li}_2\left (1-\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{2 a b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(252\) vs. \(2(37)=74\).
Time = 0.14 (sec) , antiderivative size = 252, normalized size of antiderivative = 6.81 \[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {\log ^2\left (\frac {2 a c}{(1+c) (a+b x)}\right )-2 \log (a-b x) \log \left (\frac {a+b x}{2 a}\right )+2 \log (a-b x) \log \left (\frac {a-a c+b (1+c) x}{2 a}\right )+2 \log \left (\frac {2 a c}{(1+c) (a+b x)}\right ) \log \left (-\frac {a-a c+b (1+c) x}{2 a c}\right )-2 \log (a-b x) \log \left (\frac {a-a c+b (1+c) x}{a+b x}\right )-2 \log \left (\frac {2 a c}{(1+c) (a+b x)}\right ) \log \left (\frac {a-a c+b (1+c) x}{a+b x}\right )-2 \operatorname {PolyLog}\left (2,\frac {a-b x}{2 a}\right )+2 \operatorname {PolyLog}\left (2,\frac {(1+c) (a-b x)}{2 a}\right )-2 \operatorname {PolyLog}\left (2,\frac {(1+c) (a+b x)}{2 a c}\right )}{4 a b} \]
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Time = 0.92 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\operatorname {dilog}\left (1+c -\frac {2 c a}{b x +a}\right )}{2 b a}\) | \(24\) |
default | \(\frac {\operatorname {dilog}\left (1+c -\frac {2 c a}{b x +a}\right )}{2 b a}\) | \(24\) |
risch | \(\frac {\operatorname {dilog}\left (1+c -\frac {2 c a}{b x +a}\right )}{2 b a}\) | \(24\) |
parts | \(\frac {\ln \left (\frac {a \left (1-c \right )+b \left (1+c \right ) x}{b x +a}\right ) \ln \left (b x +a \right )}{2 a b}-\frac {\ln \left (\frac {a \left (1-c \right )+b \left (1+c \right ) x}{b x +a}\right ) \ln \left (-b x +a \right )}{2 a b}+\frac {c \left (\frac {\left (1+c \right ) \left (\frac {\operatorname {dilog}\left (-\frac {\left (1+c \right ) \left (-b x +a \right )-2 a}{2 a}\right )}{1+c}+\frac {\ln \left (-b x +a \right ) \ln \left (-\frac {\left (1+c \right ) \left (-b x +a \right )-2 a}{2 a}\right )}{1+c}\right )}{2 c a}-\frac {\operatorname {dilog}\left (-\frac {-b x -a}{2 a}\right )+\ln \left (-b x +a \right ) \ln \left (-\frac {-b x -a}{2 a}\right )}{2 c a}\right )}{b}+\frac {c \left (\frac {\ln \left (b x +a \right )^{2}}{4 c a}+\frac {\left (-1-c \right ) \left (\frac {\operatorname {dilog}\left (-\frac {\left (1+c \right ) \left (b x +a \right )-2 c a}{2 c a}\right )}{1+c}+\frac {\ln \left (b x +a \right ) \ln \left (-\frac {\left (1+c \right ) \left (b x +a \right )-2 c a}{2 c a}\right )}{1+c}\right )}{2 c a}\right )}{b}\) | \(299\) |
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Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {{\rm Li}_2\left (\frac {a c - {\left (b c + b\right )} x - a}{b x + a} + 1\right )}{2 \, a b} \]
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Timed out. \[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (33) = 66\).
Time = 0.21 (sec) , antiderivative size = 246, normalized size of antiderivative = 6.65 \[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\frac {1}{2} \, {\left (\frac {\log \left (b x + a\right )}{a b} - \frac {\log \left (b x - a\right )}{a b}\right )} \log \left (\frac {b {\left (c + 1\right )} x - a {\left (c - 1\right )}}{b x + a}\right ) + \frac {\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (b x - a\right )}{4 \, a b} + \frac {\log \left (b x - a\right ) \log \left (\frac {b {\left (c + 1\right )} x - a {\left (c + 1\right )}}{2 \, a} + 1\right ) + {\rm Li}_2\left (-\frac {b {\left (c + 1\right )} x - a {\left (c + 1\right )}}{2 \, a}\right )}{2 \, a b} + \frac {\log \left (b x + a\right ) \log \left (-\frac {b x + a}{2 \, a} + 1\right ) + {\rm Li}_2\left (\frac {b x + a}{2 \, a}\right )}{2 \, a b} - \frac {\log \left (b x + a\right ) \log \left (-\frac {b {\left (c + 1\right )} x + a {\left (c + 1\right )}}{2 \, a c} + 1\right ) + {\rm Li}_2\left (\frac {b {\left (c + 1\right )} x + a {\left (c + 1\right )}}{2 \, a c}\right )}{2 \, a b} \]
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\[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\int { -\frac {\log \left (\frac {b {\left (c + 1\right )} x - a {\left (c - 1\right )}}{b x + a}\right )}{b^{2} x^{2} - a^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (\frac {a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx=\int \frac {\ln \left (-\frac {a\,\left (c-1\right )-b\,x\,\left (c+1\right )}{a+b\,x}\right )}{a^2-b^2\,x^2} \,d x \]
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