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 = 1.04, 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 12
Rule 29
Rule 30
Rule 31
Rule 36
Rule 43
Rule 193
Rule 260
Rule 263
Rule 2301
Rule 2314
Rule 2315
Rule 2316
Rule 2344
Rule 2347
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2398
Rule 2410
Rule 2411
Rule 2416
Rule 2448
Rule 2454
Rule 2455
Rule 2462
Rule 2466
Rule 2557
Rule 6099
Rule 6742
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.64, size = 111, normalized size = 1.34 \[ a b c x + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{2} \, {\left (a b - b^{2}\right )} c^{2} \log \left (c + x\right ) + \frac {1}{2} \, {\left (a b + b^{2}\right )} c^{2} \log \left (-c + x\right ) - \frac {1}{8} \, {\left (b^{2} c^{2} - b^{2} x^{2}\right )} \log \left (-\frac {c + x}{c - x}\right )^{2} + \frac {1}{2} \, {\left (b^{2} c x + a b x^{2}\right )} \log \left (-\frac {c + x}{c - x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 268, normalized size = 3.23 \[ -\frac {2 \, b^{2} c^{3} \log \left (-\frac {c + x}{c - x} - 1\right ) - 2 \, b^{2} c^{3} \log \left (-\frac {c + x}{c - x}\right ) + \frac {b^{2} {\left (c + x\right )} c^{3} \log \left (-\frac {c + x}{c - x}\right )^{2}}{{\left (c - x\right )} {\left (\frac {{\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {2 \, {\left (c + x\right )}}{c - x} + 1\right )}} + \frac {2 \, {\left (b^{2} c^{3} + \frac {2 \, a b {\left (c + x\right )} c^{3}}{c - x} + \frac {b^{2} {\left (c + x\right )} c^{3}}{c - x}\right )} \log \left (-\frac {c + x}{c - x}\right )}{\frac {{\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {2 \, {\left (c + x\right )}}{c - x} + 1} + \frac {4 \, {\left (a b c^{3} + \frac {a^{2} {\left (c + x\right )} c^{3}}{c - x} + \frac {a b {\left (c + x\right )} c^{3}}{c - x}\right )}}{\frac {{\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {2 \, {\left (c + x\right )}}{c - x} + 1}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 287, normalized size = 3.46 \[ \frac {a^{2} x^{2}}{2}+\frac {b^{2} x^{2} \arctanh \left (\frac {c}{x}\right )^{2}}{2}+c \,b^{2} \arctanh \left (\frac {c}{x}\right ) x +\frac {c^{2} b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{2}-\frac {c^{2} b^{2} \arctanh \left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{2}+\frac {c^{2} b^{2} \ln \left (\frac {c}{x}-1\right )^{2}}{8}-\frac {c^{2} b^{2} \ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {1}{2}+\frac {c}{2 x}\right )}{4}-c^{2} b^{2} \ln \left (\frac {c}{x}\right )+\frac {c^{2} b^{2} \ln \left (\frac {c}{x}-1\right )}{2}+\frac {c^{2} b^{2} \ln \left (1+\frac {c}{x}\right )}{2}+\frac {c^{2} b^{2} \ln \left (1+\frac {c}{x}\right )^{2}}{8}-\frac {c^{2} b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (1+\frac {c}{x}\right )}{4}+\frac {c^{2} b^{2} \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c}{2 x}\right )}{4}+a b \,x^{2} \arctanh \left (\frac {c}{x}\right )+a b c x +\frac {c^{2} a b \ln \left (\frac {c}{x}-1\right )}{2}-\frac {c^{2} a b \ln \left (1+\frac {c}{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 136, normalized size = 1.64 \[ \frac {1}{2} \, b^{2} x^{2} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (\frac {c}{x}\right ) - {\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c\right )} a b + \frac {1}{8} \, {\left ({\left (\log \left (c + x\right )^{2} - 2 \, {\left (\log \left (c + x\right ) - 2\right )} \log \left (-c + x\right ) + \log \left (-c + x\right )^{2} + 4 \, \log \left (c + x\right )\right )} c^{2} - 4 \, {\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c \operatorname {artanh}\left (\frac {c}{x}\right )\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 101, normalized size = 1.22 \[ \frac {a^2\,x^2}{2}-\frac {b^2\,c^2\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{2}+\frac {b^2\,x^2\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{2}+\frac {b^2\,c^2\,\ln \left (x^2-c^2\right )}{2}-a\,b\,c^2\,\mathrm {atanh}\left (\frac {c}{x}\right )+a\,b\,x^2\,\mathrm {atanh}\left (\frac {c}{x}\right )+b^2\,c\,x\,\mathrm {atanh}\left (\frac {c}{x}\right )+a\,b\,c\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 104, normalized size = 1.25 \[ \frac {a^{2} x^{2}}{2} - a b c^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )} + a b c x + a b x^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )} + b^{2} c^{2} \log {\left (- c + x \right )} - \frac {b^{2} c^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{2} + b^{2} c^{2} \operatorname {atanh}{\left (\frac {c}{x} \right )} + b^{2} c x \operatorname {atanh}{\left (\frac {c}{x} \right )} + \frac {b^{2} x^{2} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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