Optimal. Leaf size=83 \[ -\frac{1}{2} c^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{2} x^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+b c x \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{2} b^2 c^2 \log \left (1-\frac{c^2}{x^2}\right )+b^2 c^2 \log (x) \]
[Out]
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Rubi [C] time = 1.04099, antiderivative size = 574, normalized size of antiderivative = 6.92, number of steps used = 58, number of rules used = 32, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.286, Rules used = {6099, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2455, 193, 43, 6742, 30, 2557, 12, 2466, 2448, 263, 2462, 260, 2416, 2394, 2393, 2391, 2410, 2395, 36, 29, 2390} \[ -\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,\frac{c-x}{2 c}\right )-\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,-\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,\frac{c+x}{2 c}\right )+\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,1-\frac{x}{c}\right )+\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,\frac{x}{c}+1\right )+\frac{1}{2} a b c^2 \log (x)-\frac{1}{2} a b c^2 \log (c+x)-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} a b x^2 \log \left (\frac{c}{x}+1\right )+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} a b c x+\frac{1}{4} b c x \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c^2 \log (c-x)+\frac{1}{4} b^2 c^2 \log \left (\frac{c}{x}+1\right ) \log (c-x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{4} b^2 c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )+\frac{1}{4} b^2 c^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (\frac{c}{x}+1\right )-\frac{1}{4} b^2 c x \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c}{x}+1\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6099
Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2316
Rule 2315
Rule 2314
Rule 31
Rule 2455
Rule 193
Rule 43
Rule 6742
Rule 30
Rule 2557
Rule 12
Rule 2466
Rule 2448
Rule 263
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rule 2410
Rule 2395
Rule 36
Rule 29
Rule 2390
Rubi steps
\begin{align*} \int x \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} b x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 x \log ^2\left (1+\frac{c}{x}\right )\right ) \, dx\\ &=\frac{1}{4} \int x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \, dx+\frac{1}{2} b \int x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{4} b^2 \int x \log ^2\left (1+\frac{c}{x}\right ) \, dx\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{x^3} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (2 a x \log \left (1+\frac{c}{x}\right )-b x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )\right ) \, dx-\frac{1}{4} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )+(a b) \int x \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{2} b^2 \int x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{x^2 (1-c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2 (1+c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{2} b^2 \int \frac{c x \log \left (1-\frac{c}{x}\right )}{2 (-c-x)} \, dx+\frac{1}{2} b^2 \int \frac{c x \log \left (1+\frac{c}{x}\right )}{-2 c+2 x} \, dx+\frac{1}{2} (a b c) \int \frac{1}{1+\frac{c}{x}} \, dx-\frac{1}{4} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^2}-\frac{c \log (1+c x)}{x}+\frac{c^2 \log (1+c x)}{1+c x}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{2} (a b c) \int \frac{x}{c+x} \, dx+\frac{1}{4} \left (b^2 c\right ) \int \frac{x \log \left (1-\frac{c}{x}\right )}{-c-x} \, dx-\frac{1}{4} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2} \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (b^2 c\right ) \int \frac{x \log \left (1+\frac{c}{x}\right )}{-2 c+2 x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{2} (a b c) \int \left (1-\frac{c}{c+x}\right ) \, dx+\frac{1}{4} \left (b^2 c\right ) \int \left (-\log \left (1-\frac{c}{x}\right )+\frac{c \log \left (1-\frac{c}{x}\right )}{c+x}\right ) \, dx+\frac{1}{4} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{2} \left (b^2 c\right ) \int \left (\frac{1}{2} \log \left (1+\frac{c}{x}\right )-\frac{c \log \left (1+\frac{c}{x}\right )}{2 (c-x)}\right ) \, dx+\frac{1}{4} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{c}{x}\right )\\ &=\frac{1}{2} a b c x+\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (x)-\frac{1}{2} a b c^2 \log (c+x)+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} \left (b^2 c\right ) \int \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{4} \left (b^2 c\right ) \int \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{4} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{c+x} \, dx-\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{c-x} \, dx-\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a b c x-\frac{1}{4} b^2 c x \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} b^2 c x \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 c^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{2} a b c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{\left (1-\frac{c}{x}\right ) x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{\left (1+\frac{c}{x}\right ) x} \, dx+\frac{1}{4} \left (b^2 c^3\right ) \int \frac{\log (c-x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx-\frac{1}{4} \left (b^2 c^3\right ) \int \frac{\log (c+x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx\\ &=\frac{1}{2} a b c x-\frac{1}{4} b^2 c x \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} b^2 c x \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 c^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{2} a b c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{-c+x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{c+x} \, dx+\frac{1}{4} \left (b^2 c^3\right ) \int \left (\frac{\log (c-x)}{c x}-\frac{\log (c-x)}{c (c+x)}\right ) \, dx-\frac{1}{4} \left (b^2 c^3\right ) \int \left (-\frac{\log (c+x)}{c (c-x)}-\frac{\log (c+x)}{c x}\right ) \, dx\\ &=\frac{1}{2} a b c x-\frac{1}{4} b^2 c x \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} b^2 c x \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 c^2 \log (c-x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{2} a b c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log (c-x)}{x} \, dx-\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log (c-x)}{c+x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log (c+x)}{c-x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log (c+x)}{x} \, dx\\ &=\frac{1}{2} a b c x-\frac{1}{4} b^2 c x \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} b^2 c x \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 c^2 \log (c-x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{x}{c}\right )-\frac{1}{2} a b c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )+\frac{1}{4} b^2 c^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log \left (-\frac{-c-x}{2 c}\right )}{c-x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log \left (\frac{c-x}{2 c}\right )}{c+x} \, dx-\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log \left (-\frac{x}{c}\right )}{c+x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log \left (\frac{x}{c}\right )}{c-x} \, dx\\ &=\frac{1}{2} a b c x-\frac{1}{4} b^2 c x \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} b^2 c x \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 c^2 \log (c-x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{x}{c}\right )-\frac{1}{2} a b c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )+\frac{1}{4} b^2 c^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1-\frac{x}{c}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1+\frac{x}{c}\right )+\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-x\right )+\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+x\right )\\ &=\frac{1}{2} a b c x-\frac{1}{4} b^2 c x \log \left (1-\frac{c}{x}\right )+\frac{1}{4} b c \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{8} c^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{4} b^2 c x \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b x^2 \log \left (1+\frac{c}{x}\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} b^2 c^2 \log (c-x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{x}{c}\right )-\frac{1}{2} a b c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)+\frac{1}{4} b^2 c^2 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{4} b^2 c^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )+\frac{1}{4} b^2 c^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c x \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{8} b^2 x^2 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c-x}{2 c}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c+x}{2 c}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1-\frac{x}{c}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1+\frac{x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0472649, size = 92, normalized size = 1.11 \[ \frac{1}{2} \left (a^2 x^2+b c^2 (a+b) \log (x-c)-a b c^2 \log (c+x)+2 a b c x+2 b x \tanh ^{-1}\left (\frac{c}{x}\right ) (a x+b c)+b^2 \left (x^2-c^2\right ) \tanh ^{-1}\left (\frac{c}{x}\right )^2+b^2 c^2 \log (c+x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 287, normalized size = 3.5 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{{b}^{2}{x}^{2}}{2} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}+c{b}^{2}x{\it Artanh} \left ({\frac{c}{x}} \right ) +{\frac{{b}^{2}{c}^{2}}{2}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ({\frac{c}{x}}-1 \right ) }-{\frac{{b}^{2}{c}^{2}}{2}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }-{\frac{{b}^{2}{c}^{2}}{4}\ln \left ({\frac{c}{x}}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }+{\frac{{b}^{2}{c}^{2}}{8} \left ( \ln \left ({\frac{c}{x}}-1 \right ) \right ) ^{2}}+{\frac{{b}^{2}{c}^{2}}{2}\ln \left ({\frac{c}{x}}-1 \right ) }-{c}^{2}{b}^{2}\ln \left ({\frac{c}{x}} \right ) +{\frac{{b}^{2}{c}^{2}}{2}\ln \left ( 1+{\frac{c}{x}} \right ) }-{\frac{{b}^{2}{c}^{2}}{4}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{{b}^{2}{c}^{2}}{4}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }+{\frac{{b}^{2}{c}^{2}}{8} \left ( \ln \left ( 1+{\frac{c}{x}} \right ) \right ) ^{2}}+ab{x}^{2}{\it Artanh} \left ({\frac{c}{x}} \right ) +abcx+{\frac{ba{c}^{2}}{2}\ln \left ({\frac{c}{x}}-1 \right ) }-{\frac{ba{c}^{2}}{2}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975595, size = 184, normalized size = 2.22 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \operatorname{artanh}\left (\frac{c}{x}\right )^{2} + \frac{1}{2} \, a^{2} x^{2} + \frac{1}{2} \,{\left (2 \, x^{2} \operatorname{artanh}\left (\frac{c}{x}\right ) -{\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c\right )} a b + \frac{1}{8} \,{\left ({\left (\log \left (c + x\right )^{2} - 2 \,{\left (\log \left (c + x\right ) - 2\right )} \log \left (-c + x\right ) + \log \left (-c + x\right )^{2} + 4 \, \log \left (c + x\right )\right )} c^{2} - 4 \,{\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c \operatorname{artanh}\left (\frac{c}{x}\right )\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76187, size = 254, normalized size = 3.06 \begin{align*} a b c x + \frac{1}{2} \, a^{2} x^{2} - \frac{1}{2} \,{\left (a b - b^{2}\right )} c^{2} \log \left (c + x\right ) + \frac{1}{2} \,{\left (a b + b^{2}\right )} c^{2} \log \left (-c + x\right ) - \frac{1}{8} \,{\left (b^{2} c^{2} - b^{2} x^{2}\right )} \log \left (-\frac{c + x}{c - x}\right )^{2} + \frac{1}{2} \,{\left (b^{2} c x + a b x^{2}\right )} \log \left (-\frac{c + x}{c - x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05, size = 104, normalized size = 1.25 \begin{align*} \frac{a^{2} x^{2}}{2} - a b c^{2} \operatorname{atanh}{\left (\frac{c}{x} \right )} + a b c x + a b x^{2} \operatorname{atanh}{\left (\frac{c}{x} \right )} + b^{2} c^{2} \log{\left (- c + x \right )} - \frac{b^{2} c^{2} \operatorname{atanh}^{2}{\left (\frac{c}{x} \right )}}{2} + b^{2} c^{2} \operatorname{atanh}{\left (\frac{c}{x} \right )} + b^{2} c x \operatorname{atanh}{\left (\frac{c}{x} \right )} + \frac{b^{2} x^{2} \operatorname{atanh}^{2}{\left (\frac{c}{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21089, size = 159, normalized size = 1.92 \begin{align*} a b c x + \frac{1}{2} \, a^{2} x^{2} - \frac{1}{8} \,{\left (b^{2} c^{2} - b^{2} x^{2}\right )} \log \left (-\frac{c + x}{c - x}\right )^{2} - \frac{1}{2} \,{\left (a b c^{2} - b^{2} c^{2}\right )} \log \left (c + x\right ) + \frac{1}{2} \,{\left (a b c^{2} + b^{2} c^{2}\right )} \log \left (c - x\right ) + \frac{1}{2} \,{\left (b^{2} c x + a b x^{2}\right )} \log \left (-\frac{c + x}{c - x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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