Optimal. Leaf size=82 \[ \frac{1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+b c x \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{2} b^2 c^2 \log \left (\frac{c^2}{x^2}+1\right )+b^2 c^2 \log (x) \]
[Out]
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Rubi [C] time = 1.12636, antiderivative size = 663, normalized size of antiderivative = 8.09, 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 = {5035, 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} \[ -\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,\frac{c-i x}{2 c}\right )-\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,\frac{c+i x}{2 c}\right )-\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,-\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,\frac{i c}{x}\right )+\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,1-\frac{i x}{c}\right )+\frac{1}{4} b^2 c^2 \text{PolyLog}\left (2,1+\frac{i x}{c}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{2} a b c x+\frac{1}{4} b c x \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 c^2 \log \left (-\frac{c}{x}+i\right )+\frac{1}{4} b^2 c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c+i x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )+\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{i x}{c}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} i b^2 c x \log \left (1-\frac{i c}{x}\right )-\frac{1}{2} i b^2 c x \log \left (1+\frac{i c}{x}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 5035
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 \tan ^{-1}\left (\frac{c}{x}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{2} b x \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{4} b^2 x \log ^2\left (1+\frac{i c}{x}\right )\right ) \, dx\\ &=\frac{1}{4} \int x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \, dx+\frac{1}{2} b \int x \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{4} b^2 \int x \log ^2\left (1+\frac{i c}{x}\right ) \, dx\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^2}{x^3} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (-2 i a x \log \left (1+\frac{i c}{x}\right )+b x \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )\right ) \, dx+\frac{1}{4} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-(i a b) \int x \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} b^2 \int x \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (1-i c x)}{x^2 (1-i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^2 (1+i c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} (i b) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{2} b^2 \int \frac{c x \log \left (1-\frac{i c}{x}\right )}{2 (-c+i x)} \, dx-\frac{1}{2} b^2 \int \frac{c x \log \left (1+\frac{i c}{x}\right )}{2 (-c-i x)} \, dx+\frac{1}{2} (a b c) \int \frac{1}{1+\frac{i c}{x}} \, dx+\frac{1}{4} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{x^2}-\frac{i c \log (1+i c x)}{x}+\frac{i c^2 \log (1+i c x)}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} (i b) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{2} (a b c) \int \frac{x}{i c+x} \, dx+\frac{1}{4} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (b^2 c\right ) \int \frac{x \log \left (1-\frac{i c}{x}\right )}{-c+i x} \, dx-\frac{1}{4} \left (b^2 c\right ) \int \frac{x \log \left (1+\frac{i c}{x}\right )}{-c-i x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i 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+i c x)}{-i+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{4} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{2} (a b c) \int \left (1-\frac{c}{c-i x}\right ) \, dx-\frac{1}{4} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{4} \left (b^2 c\right ) \int \left (-i \log \left (1-\frac{i c}{x}\right )+\frac{i c \log \left (1-\frac{i c}{x}\right )}{c-i x}\right ) \, dx-\frac{1}{4} \left (b^2 c\right ) \int \left (i \log \left (1+\frac{i c}{x}\right )-\frac{i c \log \left (1+\frac{i c}{x}\right )}{c+i x}\right ) \, dx+\frac{1}{4} \left (i b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+i 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{i c}{x}\right )\\ &=\frac{1}{2} a b c x+\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{4} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (x)-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{4} \left (i b^2 c\right ) \int \log \left (1-\frac{i c}{x}\right ) \, dx-\frac{1}{4} \left (i b^2 c\right ) \int \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{4} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{c-i x} \, dx+\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{c+i 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 (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+i c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a b c x+\frac{1}{4} b^2 c^2 \log \left (i-\frac{c}{x}\right )+\frac{1}{4} i b^2 c x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{\left (1-\frac{i c}{x}\right ) x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{\left (1+\frac{i c}{x}\right ) x} \, dx-\frac{1}{4} \left (i b^2 c^3\right ) \int \frac{\log (c-i x)}{\left (1-\frac{i c}{x}\right ) x^2} \, dx+\frac{1}{4} \left (i b^2 c^3\right ) \int \frac{\log (c+i x)}{\left (1+\frac{i c}{x}\right ) x^2} \, dx\\ &=\frac{1}{2} a b c x+\frac{1}{4} b^2 c^2 \log \left (i-\frac{c}{x}\right )+\frac{1}{4} i b^2 c x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{-i c+x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{1}{i c+x} \, dx-\frac{1}{4} \left (i b^2 c^3\right ) \int \left (\frac{\log (c-i x)}{c (c+i x)}+\frac{i \log (c-i x)}{c x}\right ) \, dx+\frac{1}{4} \left (i b^2 c^3\right ) \int \left (\frac{\log (c+i x)}{c (c-i x)}-\frac{i \log (c+i x)}{c x}\right ) \, dx\\ &=\frac{1}{2} a b c x+\frac{1}{4} b^2 c^2 \log \left (i-\frac{c}{x}\right )+\frac{1}{4} i b^2 c x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c+i x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log (c-i x)}{c+i x} \, dx+\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log (c+i x)}{c-i x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log (c-i x)}{x} \, dx+\frac{1}{4} \left (b^2 c^2\right ) \int \frac{\log (c+i x)}{x} \, dx\\ &=\frac{1}{2} a b c x+\frac{1}{4} b^2 c^2 \log \left (i-\frac{c}{x}\right )+\frac{1}{4} i b^2 c x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c+i x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )+\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{i x}{c}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log \left (\frac{c-i x}{2 c}\right )}{c+i x} \, dx-\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log \left (\frac{c+i x}{2 c}\right )}{c-i x} \, dx-\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log \left (-\frac{i x}{c}\right )}{c+i x} \, dx+\frac{1}{4} \left (i b^2 c^2\right ) \int \frac{\log \left (\frac{i x}{c}\right )}{c-i x} \, dx\\ &=\frac{1}{2} a b c x+\frac{1}{4} b^2 c^2 \log \left (i-\frac{c}{x}\right )+\frac{1}{4} i b^2 c x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c+i x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )+\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{i x}{c}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1-\frac{i x}{c}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1+\frac{i 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-i 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+i x\right )\\ &=\frac{1}{2} a b c x+\frac{1}{4} b^2 c^2 \log \left (i-\frac{c}{x}\right )+\frac{1}{4} i b^2 c x \log \left (1-\frac{i c}{x}\right )+\frac{1}{4} b c \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{8} c^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{8} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{2} i b^2 c x \log \left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{4} b^2 x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 c^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{8} b^2 x^2 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{2} i a b c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c-i x)+\frac{1}{4} b^2 c^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{4} b^2 c^2 \log (c+i x)+\frac{1}{4} b^2 c^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)-\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{1}{2} i a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )+\frac{1}{4} b^2 c^2 \log (c-i x) \log \left (\frac{i x}{c}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c-i x}{2 c}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{c+i x}{2 c}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{4} b^2 c^2 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1-\frac{i x}{c}\right )+\frac{1}{4} b^2 c^2 \text{Li}_2\left (1+\frac{i x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0535183, size = 73, normalized size = 0.89 \[ \frac{1}{2} \left (2 b \tan ^{-1}\left (\frac{c}{x}\right ) \left (a \left (c^2+x^2\right )+b c x\right )+a x (a x+2 b c)+b^2 c^2 \log \left (c^2+x^2\right )+b^2 \left (c^2+x^2\right ) \tan ^{-1}\left (\frac{c}{x}\right )^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 116, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{{b}^{2}{x}^{2}}{2} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}+{\frac{{c}^{2}{b}^{2}}{2} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}+c{b}^{2}\arctan \left ({\frac{c}{x}} \right ) x+{\frac{{c}^{2}{b}^{2}}{2}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }-{c}^{2}{b}^{2}\ln \left ({\frac{c}{x}} \right ) +ab{x}^{2}\arctan \left ({\frac{c}{x}} \right ) -{c}^{2}ab\arctan \left ({\frac{x}{c}} \right ) +abcx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5402, size = 136, normalized size = 1.66 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \arctan \left (\frac{c}{x}\right )^{2} + \frac{1}{2} \, a^{2} x^{2} +{\left (x^{2} \arctan \left (\frac{c}{x}\right ) -{\left (c \arctan \left (\frac{x}{c}\right ) - x\right )} c\right )} a b - \frac{1}{2} \,{\left ({\left (\arctan \left (x, c\right )^{2} - \log \left (c^{2} + x^{2}\right )\right )} c^{2} + 2 \,{\left (c \arctan \left (\frac{x}{c}\right ) - x\right )} c \arctan \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 = 2.28212, size = 201, normalized size = 2.45 \begin{align*} -a b c^{2} \arctan \left (\frac{x}{c}\right ) + \frac{1}{2} \, b^{2} c^{2} \log \left (c^{2} + x^{2}\right ) + a b c x + \frac{1}{2} \, a^{2} x^{2} + \frac{1}{2} \,{\left (b^{2} c^{2} + b^{2} x^{2}\right )} \arctan \left (\frac{c}{x}\right )^{2} +{\left (b^{2} c x + a b x^{2}\right )} \arctan \left (\frac{c}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.586846, size = 97, normalized size = 1.18 \begin{align*} \frac{a^{2} x^{2}}{2} + a b c^{2} \operatorname{atan}{\left (\frac{c}{x} \right )} + a b c x + a b x^{2} \operatorname{atan}{\left (\frac{c}{x} \right )} + \frac{b^{2} c^{2} \log{\left (c^{2} + x^{2} \right )}}{2} + \frac{b^{2} c^{2} \operatorname{atan}^{2}{\left (\frac{c}{x} \right )}}{2} + b^{2} c x \operatorname{atan}{\left (\frac{c}{x} \right )} + \frac{b^{2} x^{2} \operatorname{atan}^{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 = 1.17458, size = 173, normalized size = 2.11 \begin{align*} \frac{1}{2} \, b^{2} c^{2} \arctan \left (\frac{c}{x}\right )^{2} + \frac{1}{2} \, b^{2} x^{2} \arctan \left (\frac{c}{x}\right )^{2} + \frac{1}{2} \, a b c^{2} i \log \left (i x + c\right ) - \frac{1}{2} \, a b c^{2} i \log \left (-i x + c\right ) + b^{2} c x \arctan \left (\frac{c}{x}\right ) + a b x^{2} \arctan \left (\frac{c}{x}\right ) + \frac{1}{2} \, b^{2} c^{2} \log \left (i x + c\right ) + \frac{1}{2} \, b^{2} c^{2} \log \left (-i x + c\right ) + a b c x + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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