Optimal. Leaf size=152 \[ -\frac{1}{3} i b^2 c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-\frac{i c}{x}}\right )-\frac{1}{3} i c^3 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{2}{3} b c^3 \log \left (2-\frac{2}{1-\frac{i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{3} b c x^2 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{3} x^3 \left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{3} b^2 c^2 x+\frac{1}{3} b^2 c^3 \cot ^{-1}\left (\frac{x}{c}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 1.49016, antiderivative size = 787, normalized size of antiderivative = 5.18, number of steps used = 73, number of rules used = 34, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.125, Rules used = {5035, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 2455, 263, 43, 6742, 30, 2557, 12, 2466, 2448, 2462, 260, 2416, 2394, 2393, 2391, 193, 2410, 2395, 36, 29, 2390} \[ \frac{1}{6} i b^2 c^3 \text{PolyLog}\left (2,\frac{c-i x}{2 c}\right )-\frac{1}{6} i b^2 c^3 \text{PolyLog}\left (2,\frac{c+i x}{2 c}\right )+\frac{1}{6} i b^2 c^3 \text{PolyLog}\left (2,-\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{PolyLog}\left (2,\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{PolyLog}\left (2,1-\frac{i x}{c}\right )+\frac{1}{6} i b^2 c^3 \text{PolyLog}\left (2,1+\frac{i x}{c}\right )-\frac{1}{3} i a b c^2 x+\frac{1}{12} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{3} a b c^3 \log (c-i x)-\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} i b c^2 x \left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{6} a b c x^2+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{3} b^2 c^2 x+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{4} i b^2 c^3 \log \left (-\frac{c}{x}+i\right )+\frac{1}{12} i b^2 c^3 \log \left (\frac{c}{x}+i\right )+\frac{1}{12} i b^2 c^3 \log (c-i x)-\frac{1}{6} i b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{12} i b^2 c^3 \log (c+i x)+\frac{1}{6} i b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{6} i b^2 c^3 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )+\frac{1}{6} i b^2 c^3 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c x^2 \log \left (1-\frac{i c}{x}\right )-\frac{1}{6} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \]
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 2319
Rule 44
Rule 2455
Rule 263
Rule 43
Rule 6742
Rule 30
Rule 2557
Rule 12
Rule 2466
Rule 2448
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rule 193
Rule 2410
Rule 2395
Rule 36
Rule 29
Rule 2390
Rubi steps
\begin{align*} \int x^2 \left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{2} b x^2 \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^2 \log ^2\left (1+\frac{i c}{x}\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2 \, dx+\frac{1}{2} b \int x^2 \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^2 \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^4} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (-2 i a x^2 \log \left (1+\frac{i c}{x}\right )+b x^2 \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^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-(i a b) \int x^2 \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{2} b^2 \int x^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right ) \, dx-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (1-i c x)}{x^3 (1-i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^3 (1+i c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{6} (i b) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^3} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{2} b^2 \int \frac{c x^2 \log \left (1-\frac{i c}{x}\right )}{3 (-c+i x)} \, dx-\frac{1}{2} b^2 \int \frac{c x^2 \log \left (1+\frac{i c}{x}\right )}{3 (-c-i x)} \, dx+\frac{1}{3} (a b c) \int \frac{x}{1+\frac{i c}{x}} \, dx+\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{x^3}-\frac{i c \log (1+i c x)}{x^2}-\frac{c^2 \log (1+i c x)}{x}+\frac{c^3 \log (1+i c x)}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{6} (i b) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^3} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{6} (b c) \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}{3} (a b c) \int \frac{x^2}{i c+x} \, dx+\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^3} \, dx,x,\frac{1}{x}\right )-\frac{1}{6} \left (b^2 c\right ) \int \frac{x^2 \log \left (1-\frac{i c}{x}\right )}{-c+i x} \, dx-\frac{1}{6} \left (b^2 c\right ) \int \frac{x^2 \log \left (1+\frac{i c}{x}\right )}{-c-i x} \, dx+\frac{1}{6} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{6} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (i b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} b^2 c^2 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{6} (b c) \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}{3} (a b c) \int \left (-i c+\frac{i c^2}{c-i x}+x\right ) \, dx-\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{6} \left (b^2 c\right ) \int \left (-c \log \left (1-\frac{i c}{x}\right )+\frac{c^2 \log \left (1-\frac{i c}{x}\right )}{c-i x}-i x \log \left (1-\frac{i c}{x}\right )\right ) \, dx-\frac{1}{6} \left (b^2 c\right ) \int \left (-c \log \left (1+\frac{i c}{x}\right )+\frac{c^2 \log \left (1+\frac{i c}{x}\right )}{c+i x}+i x \log \left (1+\frac{i c}{x}\right )\right ) \, dx+\frac{1}{6} \left (i b c^2\right ) \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}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+i c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{i c}{x}\right )\\ &=-\frac{1}{3} i a b c^2 x+\frac{1}{6} a b c x^2+\frac{1}{6} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} b^2 c^2 x \log \left (1+\frac{i c}{x}\right )-\frac{1}{12} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{3} a b c^3 \log (c-i x)+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{c^2}{(-1+x)^2}+\frac{c^2}{-1+x}-\frac{c^2}{x}\right ) \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{6} \left (i b^2 c\right ) \int x \log \left (1-\frac{i c}{x}\right ) \, dx-\frac{1}{6} \left (i b^2 c\right ) \int x \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{6} \left (i b c^2\right ) \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}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{i c}{x}+\frac{i c^2}{-i+c x}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{6} \left (b^2 c^2\right ) \int \log \left (1-\frac{i c}{x}\right ) \, dx+\frac{1}{6} \left (b^2 c^2\right ) \int \log \left (1+\frac{i c}{x}\right ) \, dx+\frac{1}{6} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-\frac{i c}{x}\right )-\frac{1}{6} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{x} \, dx,x,1-\frac{i c}{x}\right )+\frac{1}{6} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{c-i x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{c+i x} \, dx+\frac{1}{6} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+i c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3} i a b c^2 x+\frac{1}{6} b^2 c^2 x+\frac{1}{6} a b c x^2-\frac{1}{4} i b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{12} i b^2 c^3 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{12} i b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{3} a b c^3 \log (c-i x)-\frac{1}{6} i b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{6} i b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{1}{1-\frac{i c}{x}} \, dx+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{1}{1+\frac{i c}{x}} \, dx-\frac{1}{6} \left (b^2 c^2\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}{6} \left (i b^2 c^3\right ) \int \frac{1}{\left (1-\frac{i c}{x}\right ) x} \, dx+\frac{1}{6} \left (i b^2 c^3\right ) \int \frac{1}{\left (1+\frac{i c}{x}\right ) x} \, dx-\frac{1}{6} \left (b^2 c^4\right ) \int \frac{\log (c-i x)}{\left (1-\frac{i c}{x}\right ) x^2} \, dx-\frac{1}{6} \left (b^2 c^4\right ) \int \frac{\log (c+i x)}{\left (1+\frac{i c}{x}\right ) x^2} \, dx\\ &=-\frac{1}{3} i a b c^2 x+\frac{1}{6} b^2 c^2 x+\frac{1}{6} a b c x^2-\frac{1}{4} i b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{12} i b^2 c^3 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{12} i b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{3} a b c^3 \log (c-i x)-\frac{1}{6} i b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)+\frac{1}{6} i b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{x}{-i c+x} \, dx+\frac{1}{12} \left (b^2 c^2\right ) \int \frac{x}{i c+x} \, dx-\frac{1}{6} \left (i b^2 c^3\right ) \int \frac{1}{-i c+x} \, dx+\frac{1}{6} \left (i b^2 c^3\right ) \int \frac{1}{i c+x} \, dx-\frac{1}{6} \left (b^2 c^4\right ) \int \left (\frac{\log (c-i x)}{c (c+i x)}+\frac{i \log (c-i x)}{c x}\right ) \, dx-\frac{1}{6} \left (b^2 c^4\right ) \int \left (\frac{\log (c+i x)}{c (c-i x)}-\frac{i \log (c+i x)}{c x}\right ) \, dx\\ &=-\frac{1}{3} i a b c^2 x+\frac{1}{6} b^2 c^2 x+\frac{1}{6} a b c x^2-\frac{1}{4} i b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{12} i b^2 c^3 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{12} i b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{3} a b c^3 \log (c-i x)+\frac{1}{6} i b^2 c^3 \log (c-i x)-\frac{1}{6} i b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{6} i b^2 c^3 \log (c+i x)+\frac{1}{6} i b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (\frac{i c}{x}\right )+\frac{1}{12} \left (b^2 c^2\right ) \int \left (1-\frac{c}{c-i x}\right ) \, dx+\frac{1}{12} \left (b^2 c^2\right ) \int \left (1-\frac{c}{c+i x}\right ) \, dx-\frac{1}{6} \left (i b^2 c^3\right ) \int \frac{\log (c-i x)}{x} \, dx+\frac{1}{6} \left (i b^2 c^3\right ) \int \frac{\log (c+i x)}{x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log (c-i x)}{c+i x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log (c+i x)}{c-i x} \, dx\\ &=-\frac{1}{3} i a b c^2 x+\frac{1}{3} b^2 c^2 x+\frac{1}{6} a b c x^2-\frac{1}{4} i b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{12} i b^2 c^3 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{12} i b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{3} a b c^3 \log (c-i x)+\frac{1}{12} i b^2 c^3 \log (c-i x)-\frac{1}{6} i b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{12} i b^2 c^3 \log (c+i x)+\frac{1}{6} i b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{6} i b^2 c^3 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} i b^2 c^3 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (\frac{c-i x}{2 c}\right )}{c+i x} \, dx-\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (\frac{c+i x}{2 c}\right )}{c-i x} \, dx+\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (-\frac{i x}{c}\right )}{c+i x} \, dx+\frac{1}{6} \left (b^2 c^3\right ) \int \frac{\log \left (\frac{i x}{c}\right )}{c-i x} \, dx\\ &=-\frac{1}{3} i a b c^2 x+\frac{1}{3} b^2 c^2 x+\frac{1}{6} a b c x^2-\frac{1}{4} i b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{12} i b^2 c^3 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{12} i b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{3} a b c^3 \log (c-i x)+\frac{1}{12} i b^2 c^3 \log (c-i x)-\frac{1}{6} i b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{12} i b^2 c^3 \log (c+i x)+\frac{1}{6} i b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{6} i b^2 c^3 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} i b^2 c^3 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (1-\frac{i x}{c}\right )+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (1+\frac{i x}{c}\right )-\frac{1}{6} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-i x\right )+\frac{1}{6} \left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+i x\right )\\ &=-\frac{1}{3} i a b c^2 x+\frac{1}{3} b^2 c^2 x+\frac{1}{6} a b c x^2-\frac{1}{4} i b^2 c^3 \log \left (i-\frac{c}{x}\right )+\frac{1}{12} i b^2 c^3 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} b^2 c^2 x \log \left (1-\frac{i c}{x}\right )+\frac{1}{12} i b^2 c x^2 \log \left (1-\frac{i c}{x}\right )+\frac{1}{6} i b c^2 \left (1-\frac{i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )+\frac{1}{12} i c^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2-\frac{1}{6} i b^2 c x^2 \log \left (1+\frac{i c}{x}\right )-\frac{1}{3} i a b x^3 \log \left (1+\frac{i c}{x}\right )+\frac{1}{6} b^2 x^3 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )+\frac{1}{12} i b^2 c^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+\frac{i c}{x}\right )-\frac{1}{3} a b c^3 \log (c-i x)+\frac{1}{12} i b^2 c^3 \log (c-i x)-\frac{1}{6} i b^2 c^3 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)-\frac{1}{12} i b^2 c^3 \log (c+i x)+\frac{1}{6} i b^2 c^3 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)-\frac{1}{6} i b^2 c^3 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)+\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )-\frac{1}{3} a b c^3 \log (x)+\frac{1}{6} i b^2 c^3 \log (c+i x) \log \left (-\frac{i x}{c}\right )-\frac{1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac{i x}{c}\right )+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (\frac{c-i x}{2 c}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (\frac{c+i x}{2 c}\right )+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (-\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (\frac{i c}{x}\right )-\frac{1}{6} i b^2 c^3 \text{Li}_2\left (1-\frac{i x}{c}\right )+\frac{1}{6} i b^2 c^3 \text{Li}_2\left (1+\frac{i x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.328369, size = 152, normalized size = 1. \[ \frac{1}{3} \left (-i b^2 c^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )+a^2 x^3-a b c^3 \log \left (\frac{c^2}{x^2}+1\right )+b \tan ^{-1}\left (\frac{c}{x}\right ) \left (2 a x^3+b c \left (c^2+x^2\right )+2 b c^3 \log \left (1-e^{2 i \tan ^{-1}\left (\frac{c}{x}\right )}\right )\right )+2 a b c^3 \log \left (\frac{c}{x}\right )+a b c x^2+b^2 \left (x^3-i c^3\right ) \tan ^{-1}\left (\frac{c}{x}\right )^2+b^2 c^2 x\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.104, size = 445, normalized size = 2.9 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{{b}^{2}{x}^{3}}{3} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{{c}^{3}{b}^{2}}{3}\arctan \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }+{\frac{{b}^{2}c{x}^{2}}{3}\arctan \left ({\frac{c}{x}} \right ) }+{\frac{2\,{c}^{3}{b}^{2}}{3}\ln \left ({\frac{c}{x}} \right ) \arctan \left ({\frac{c}{x}} \right ) }+{\frac{i}{6}}{c}^{3}{b}^{2}\ln \left ({\frac{c}{x}}-i \right ) \ln \left ( -{\frac{i}{2}} \left ({\frac{c}{x}}+i \right ) \right ) -{\frac{i}{12}}{c}^{3}{b}^{2} \left ( \ln \left ({\frac{c}{x}}+i \right ) \right ) ^{2}-{\frac{i}{6}}{c}^{3}{b}^{2}{\it dilog} \left ({\frac{i}{2}} \left ({\frac{c}{x}}-i \right ) \right ) +{\frac{i}{3}}{c}^{3}{b}^{2}{\it dilog} \left ( 1+{\frac{ic}{x}} \right ) -{\frac{i}{6}}{c}^{3}{b}^{2}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) \ln \left ({\frac{c}{x}}-i \right ) -{\frac{i}{3}}{c}^{3}{b}^{2}\ln \left ({\frac{c}{x}} \right ) \ln \left ( 1-{\frac{ic}{x}} \right ) +{\frac{i}{3}}{c}^{3}{b}^{2}\ln \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{ic}{x}} \right ) +{\frac{i}{6}}{c}^{3}{b}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ({\frac{c}{x}}+i \right ) \right ) -{\frac{{c}^{3}{b}^{2}}{3}\arctan \left ({\frac{x}{c}} \right ) }+{\frac{{b}^{2}{c}^{2}x}{3}}+{\frac{i}{6}}{c}^{3}{b}^{2}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) \ln \left ({\frac{c}{x}}+i \right ) +{\frac{i}{12}}{c}^{3}{b}^{2} \left ( \ln \left ({\frac{c}{x}}-i \right ) \right ) ^{2}-{\frac{i}{6}}{c}^{3}{b}^{2}\ln \left ({\frac{c}{x}}+i \right ) \ln \left ({\frac{i}{2}} \left ({\frac{c}{x}}-i \right ) \right ) -{\frac{i}{3}}{c}^{3}{b}^{2}{\it dilog} \left ( 1-{\frac{ic}{x}} \right ) +{\frac{2\,ab{x}^{3}}{3}\arctan \left ({\frac{c}{x}} \right ) }-{\frac{a{c}^{3}b}{3}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }+{\frac{abc{x}^{2}}{3}}+{\frac{2\,a{c}^{3}b}{3}\ln \left ({\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} x^{3} + \frac{1}{3} \,{\left (2 \, x^{3} \arctan \left (\frac{c}{x}\right ) -{\left (c^{2} \log \left (c^{2} + x^{2}\right ) - x^{2}\right )} c\right )} a b + \frac{1}{48} \,{\left (4 \, x^{3} \arctan \left (c, x\right )^{2} - x^{3} \log \left (c^{2} + x^{2}\right )^{2} + 48 \, \int \frac{36 \, c^{2} x^{2} \arctan \left (c, x\right )^{2} + 36 \, x^{4} \arctan \left (c, x\right )^{2} + 8 \, c x^{3} \arctan \left (c, x\right ) + 4 \, x^{4} \log \left (c^{2} + x^{2}\right ) + 3 \,{\left (c^{2} x^{2} + x^{4}\right )} \log \left (c^{2} + x^{2}\right )^{2}}{48 \,{\left (c^{2} + x^{2}\right )}}\,{d x}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{2} \arctan \left (\frac{c}{x}\right )^{2} + 2 \, a b x^{2} \arctan \left (\frac{c}{x}\right ) + a^{2} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{atan}{\left (\frac{c}{x} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (\frac{c}{x}\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]