Optimal. Leaf size=84 \[ -\frac{\left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 c^2}-\frac{\left (a+b \cot ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 x^2}+\frac{a b}{c x}-\frac{b^2 \log \left (\frac{c^2}{x^2}+1\right )}{2 c^2}+\frac{b^2 \cot ^{-1}\left (\frac{x}{c}\right )}{c x} \]
[Out]
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Rubi [C] time = 1.29363, antiderivative size = 836, normalized size of antiderivative = 9.95, number of steps used = 66, number of rules used = 23, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.438, Rules used = {5035, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2395, 43, 6742, 30, 2557, 12, 2466, 2462, 260, 2416, 2394, 2393, 2391, 2315} \[ -\frac{\left (1-\frac{i c}{x}\right )^2 b^2}{16 c^2}-\frac{\left (\frac{i c}{x}+1\right )^2 b^2}{16 c^2}-\frac{\left (\frac{i c}{x}+1\right )^2 \log ^2\left (\frac{i c}{x}+1\right ) b^2}{8 c^2}+\frac{\left (\frac{i c}{x}+1\right ) \log ^2\left (\frac{i c}{x}+1\right ) b^2}{4 c^2}+\frac{\log \left (i-\frac{c}{x}\right ) b^2}{8 c^2}-\frac{3 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right ) b^2}{4 c^2}+\frac{\log \left (1-\frac{i c}{x}\right ) b^2}{8 x^2}+\frac{\left (\frac{i c}{x}+1\right )^2 \log \left (\frac{i c}{x}+1\right ) b^2}{8 c^2}-\frac{3 \left (\frac{i c}{x}+1\right ) \log \left (\frac{i c}{x}+1\right ) b^2}{4 c^2}-\frac{\log \left (1-\frac{i c}{x}\right ) \log \left (\frac{i c}{x}+1\right ) b^2}{4 x^2}+\frac{\log \left (\frac{i c}{x}+1\right ) b^2}{8 x^2}+\frac{\log \left (\frac{c}{x}+i\right ) b^2}{8 c^2}-\frac{\log \left (1-\frac{i c}{x}\right ) \log (c-i x) b^2}{4 c^2}-\frac{\log \left (\frac{i c}{x}+1\right ) \log (c+i x) b^2}{4 c^2}+\frac{\log \left (\frac{c-i x}{2 c}\right ) \log (c+i x) b^2}{4 c^2}+\frac{\log (c-i x) \log \left (\frac{c+i x}{2 c}\right ) b^2}{4 c^2}-\frac{\log (c+i x) \log \left (-\frac{i x}{c}\right ) b^2}{4 c^2}-\frac{\log (c-i x) \log \left (\frac{i x}{c}\right ) b^2}{4 c^2}+\frac{\text{PolyLog}\left (2,\frac{c-i x}{2 c}\right ) b^2}{4 c^2}+\frac{\text{PolyLog}\left (2,\frac{c+i x}{2 c}\right ) b^2}{4 c^2}+\frac{\text{PolyLog}\left (2,-\frac{i c}{x}\right ) b^2}{4 c^2}+\frac{\text{PolyLog}\left (2,\frac{i c}{x}\right ) b^2}{4 c^2}-\frac{\text{PolyLog}\left (2,1-\frac{i x}{c}\right ) b^2}{4 c^2}-\frac{\text{PolyLog}\left (2,\frac{i x}{c}+1\right ) b^2}{4 c^2}-\frac{b^2}{8 x^2}+\frac{i a \log \left (i-\frac{c}{x}\right ) b}{2 c^2}-\frac{i \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right ) b}{8 c^2}+\frac{i a \log \left (\frac{i c}{x}+1\right ) b}{2 x^2}+\frac{3 a b}{2 c x}-\frac{i a b}{4 x^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 5035
Rule 2454
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rule 2395
Rule 43
Rule 6742
Rule 30
Rule 2557
Rule 12
Rule 2466
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rule 2315
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-1}\left (\frac{c}{x}\right )\right )^2}{x^3} \, dx &=\int \left (\frac{\left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 x^3}+\frac{b \left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log \left (1+\frac{i c}{x}\right )}{2 x^3}-\frac{b^2 \log ^2\left (1+\frac{i c}{x}\right )}{4 x^3}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{x^3} \, dx+\frac{1}{2} b \int \frac{\left (-2 i a+b \log \left (1-\frac{i c}{x}\right )\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{4} b^2 \int \frac{\log ^2\left (1+\frac{i c}{x}\right )}{x^3} \, dx\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int x (2 a+i b \log (1-i c x))^2 \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (-\frac{2 i a \log \left (1+\frac{i c}{x}\right )}{x^3}+\frac{b \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3}\right ) \, dx+\frac{1}{4} b^2 \operatorname{Subst}\left (\int x \log ^2(1+i c x) \, dx,x,\frac{1}{x}\right )\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{i (2 a+i b \log (1-i c x))^2}{c}+\frac{i (1-i c x) (2 a+i b \log (1-i c x))^2}{c}\right ) \, dx,x,\frac{1}{x}\right )\right )-(i a b) \int \frac{\log \left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{1}{4} b^2 \operatorname{Subst}\left (\int \left (\frac{i \log ^2(1+i c x)}{c}-\frac{i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{2} b^2 \int \frac{\log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{x^3} \, dx\\ &=-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+(i a b) \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{2} b^2 \int \frac{c \log \left (1-\frac{i c}{x}\right )}{2 (c-i x) x^3} \, dx-\frac{1}{2} b^2 \int \frac{c \log \left (1+\frac{i c}{x}\right )}{2 (c+i x) x^3} \, dx+\frac{i \operatorname{Subst}\left (\int (2 a+i b \log (1-i c x))^2 \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{i \operatorname{Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^2 \, dx,x,\frac{1}{x}\right )}{4 c}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}-\frac{\operatorname{Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-\frac{i c}{x}\right )}{4 c^2}+\frac{\operatorname{Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}+\frac{1}{2} (a b c) \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (b^2 c\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{(c-i x) x^3} \, dx-\frac{1}{4} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{(c+i x) x^3} \, dx\\ &=-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{(i b) \operatorname{Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-\frac{i c}{x}\right )}{4 c^2}+\frac{(i b) \operatorname{Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-\frac{i c}{x}\right )}{2 c^2}+\frac{b^2 \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{i c}{x}\right )}{2 c^2}+\frac{1}{2} (a b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (b^2 c\right ) \int \left (-\frac{i \log \left (1-\frac{i c}{x}\right )}{c^3 (c-i x)}+\frac{\log \left (1-\frac{i c}{x}\right )}{c x^3}+\frac{i \log \left (1-\frac{i c}{x}\right )}{c^2 x^2}-\frac{\log \left (1-\frac{i c}{x}\right )}{c^3 x}\right ) \, dx-\frac{1}{4} \left (b^2 c\right ) \int \left (\frac{i \log \left (1+\frac{i c}{x}\right )}{c^3 (c+i x)}+\frac{\log \left (1+\frac{i c}{x}\right )}{c x^3}-\frac{i \log \left (1+\frac{i c}{x}\right )}{c^2 x^2}-\frac{\log \left (1+\frac{i c}{x}\right )}{c^3 x}\right ) \, dx\\ &=-\frac{b^2 \left (1-\frac{i c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{i a b}{4 x^2}+\frac{3 a b}{2 c x}+\frac{i b^2}{2 c x}+\frac{i a b \log \left (i-\frac{c}{x}\right )}{2 c^2}-\frac{i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{2 c^2}+\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{1}{4} b^2 \int \frac{\log \left (1-\frac{i c}{x}\right )}{x^3} \, dx-\frac{1}{4} b^2 \int \frac{\log \left (1+\frac{i c}{x}\right )}{x^3} \, dx+\frac{\left (i b^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{c-i x} \, dx}{4 c^2}-\frac{\left (i b^2\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{c+i x} \, dx}{4 c^2}+\frac{b^2 \int \frac{\log \left (1-\frac{i c}{x}\right )}{x} \, dx}{4 c^2}+\frac{b^2 \int \frac{\log \left (1+\frac{i c}{x}\right )}{x} \, dx}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{i c}{x}\right )}{2 c^2}-\frac{\left (i b^2\right ) \int \frac{\log \left (1-\frac{i c}{x}\right )}{x^2} \, dx}{4 c}+\frac{\left (i b^2\right ) \int \frac{\log \left (1+\frac{i c}{x}\right )}{x^2} \, dx}{4 c}\\ &=-\frac{b^2 \left (1-\frac{i c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{i a b}{4 x^2}+\frac{3 a b}{2 c x}+\frac{i a b \log \left (i-\frac{c}{x}\right )}{2 c^2}-\frac{b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{2 c^2}-\frac{i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{2 c^2}+\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}+\frac{1}{4} b^2 \operatorname{Subst}\left (\int x \log (1-i c x) \, dx,x,\frac{1}{x}\right )+\frac{1}{4} b^2 \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,\frac{1}{x}\right )+\frac{\left (i b^2\right ) \int \frac{\log (c-i x)}{\left (1-\frac{i c}{x}\right ) x^2} \, dx}{4 c}-\frac{\left (i b^2\right ) \int \frac{\log (c+i x)}{\left (1+\frac{i c}{x}\right ) x^2} \, dx}{4 c}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=-\frac{b^2 \left (1-\frac{i c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{i a b}{4 x^2}+\frac{3 a b}{2 c x}+\frac{i a b \log \left (i-\frac{c}{x}\right )}{2 c^2}-\frac{b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{2 c^2}+\frac{b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{2 c^2}+\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}+\frac{b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{i c}{x}\right )}{4 c^2}+\frac{\left (i b^2\right ) \int \left (\frac{\log (c-i x)}{c (c+i x)}+\frac{i \log (c-i x)}{c x}\right ) \, dx}{4 c}-\frac{\left (i b^2\right ) \int \left (\frac{\log (c+i x)}{c (c-i x)}-\frac{i \log (c+i x)}{c x}\right ) \, dx}{4 c}+\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b^2 \left (1-\frac{i c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{i a b}{4 x^2}+\frac{3 a b}{2 c x}+\frac{i a b \log \left (i-\frac{c}{x}\right )}{2 c^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}+\frac{b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}+\frac{\left (i b^2\right ) \int \frac{\log (c-i x)}{c+i x} \, dx}{4 c^2}-\frac{\left (i b^2\right ) \int \frac{\log (c+i x)}{c-i x} \, dx}{4 c^2}-\frac{b^2 \int \frac{\log (c-i x)}{x} \, dx}{4 c^2}-\frac{b^2 \int \frac{\log (c+i x)}{x} \, dx}{4 c^2}-\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{i x}{c}-\frac{i}{c^2 (i+c x)}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b^2 \left (1-\frac{i c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{i a b}{4 x^2}-\frac{b^2}{8 x^2}+\frac{3 a b}{2 c x}+\frac{i a b \log \left (i-\frac{c}{x}\right )}{2 c^2}+\frac{b^2 \log \left (i-\frac{c}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}+\frac{b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{b^2 \log \left (i+\frac{c}{x}\right )}{8 c^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )}{4 c^2}-\frac{b^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )}{4 c^2}-\frac{b^2 \log (c-i x) \log \left (\frac{i x}{c}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}-\frac{\left (i b^2\right ) \int \frac{\log \left (\frac{c-i x}{2 c}\right )}{c+i x} \, dx}{4 c^2}+\frac{\left (i b^2\right ) \int \frac{\log \left (\frac{c+i x}{2 c}\right )}{c-i x} \, dx}{4 c^2}+\frac{\left (i b^2\right ) \int \frac{\log \left (-\frac{i x}{c}\right )}{c+i x} \, dx}{4 c^2}-\frac{\left (i b^2\right ) \int \frac{\log \left (\frac{i x}{c}\right )}{c-i x} \, dx}{4 c^2}\\ &=-\frac{b^2 \left (1-\frac{i c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{i a b}{4 x^2}-\frac{b^2}{8 x^2}+\frac{3 a b}{2 c x}+\frac{i a b \log \left (i-\frac{c}{x}\right )}{2 c^2}+\frac{b^2 \log \left (i-\frac{c}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}+\frac{b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{b^2 \log \left (i+\frac{c}{x}\right )}{8 c^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )}{4 c^2}-\frac{b^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )}{4 c^2}-\frac{b^2 \log (c-i x) \log \left (\frac{i x}{c}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1-\frac{i x}{c}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1+\frac{i x}{c}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-i x\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+i x\right )}{4 c^2}\\ &=-\frac{b^2 \left (1-\frac{i c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2}{16 c^2}-\frac{i a b}{4 x^2}-\frac{b^2}{8 x^2}+\frac{3 a b}{2 c x}+\frac{i a b \log \left (i-\frac{c}{x}\right )}{2 c^2}+\frac{b^2 \log \left (i-\frac{c}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1-\frac{i c}{x}\right ) \log \left (1-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \log \left (1-\frac{i c}{x}\right )}{8 x^2}-\frac{i b \left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )}{8 c^2}-\frac{\left (1-\frac{i c}{x}\right ) \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{4 c^2}+\frac{\left (1-\frac{i c}{x}\right )^2 \left (2 a+i b \log \left (1-\frac{i c}{x}\right )\right )^2}{8 c^2}-\frac{3 b^2 \left (1+\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log \left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{i a b \log \left (1+\frac{i c}{x}\right )}{2 x^2}+\frac{b^2 \log \left (1+\frac{i c}{x}\right )}{8 x^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log \left (1+\frac{i c}{x}\right )}{4 x^2}+\frac{b^2 \left (1+\frac{i c}{x}\right ) \log ^2\left (1+\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{i c}{x}\right )^2 \log ^2\left (1+\frac{i c}{x}\right )}{8 c^2}+\frac{b^2 \log \left (i+\frac{c}{x}\right )}{8 c^2}-\frac{b^2 \log \left (1-\frac{i c}{x}\right ) \log (c-i x)}{4 c^2}-\frac{b^2 \log \left (1+\frac{i c}{x}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \log \left (\frac{c-i x}{2 c}\right ) \log (c+i x)}{4 c^2}+\frac{b^2 \log (c-i x) \log \left (\frac{c+i x}{2 c}\right )}{4 c^2}-\frac{b^2 \log (c+i x) \log \left (-\frac{i x}{c}\right )}{4 c^2}-\frac{b^2 \log (c-i x) \log \left (\frac{i x}{c}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c-i x}{2 c}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c+i x}{2 c}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{i c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{i c}{x}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1-\frac{i x}{c}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1+\frac{i x}{c}\right )}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.073959, size = 99, normalized size = 1.18 \[ -\frac{a^2 c^2-2 a b x^2 \tan ^{-1}\left (\frac{x}{c}\right )-2 a b c x+2 b c \tan ^{-1}\left (\frac{c}{x}\right ) (a c-b x)+b^2 x^2 \log \left (c^2+x^2\right )+b^2 \left (c^2+x^2\right ) \tan ^{-1}\left (\frac{c}{x}\right )^2-2 b^2 x^2 \log (x)}{2 c^2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 110, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{{b}^{2}}{2\,{x}^{2}} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{{b}^{2}}{2\,{c}^{2}} \left ( \arctan \left ({\frac{c}{x}} \right ) \right ) ^{2}}+{\frac{{b}^{2}}{cx}\arctan \left ({\frac{c}{x}} \right ) }-{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }-{\frac{ab}{{x}^{2}}\arctan \left ({\frac{c}{x}} \right ) }+{\frac{ab}{{c}^{2}}\arctan \left ({\frac{x}{c}} \right ) }+{\frac{ab}{cx}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62002, size = 158, normalized size = 1.88 \begin{align*}{\left (c{\left (\frac{\arctan \left (\frac{x}{c}\right )}{c^{3}} + \frac{1}{c^{2} x}\right )} - \frac{\arctan \left (\frac{c}{x}\right )}{x^{2}}\right )} a b + \frac{1}{2} \,{\left (2 \, c{\left (\frac{\arctan \left (\frac{x}{c}\right )}{c^{3}} + \frac{1}{c^{2} x}\right )} \arctan \left (\frac{c}{x}\right ) + \frac{\arctan \left (x, c\right )^{2} - \log \left (c^{2} + x^{2}\right ) + 2 \, \log \left (x\right )}{c^{2}}\right )} b^{2} - \frac{b^{2} \arctan \left (\frac{c}{x}\right )^{2}}{2 \, x^{2}} - \frac{a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31232, size = 239, normalized size = 2.85 \begin{align*} \frac{2 \, a b x^{2} \arctan \left (\frac{x}{c}\right ) - b^{2} x^{2} \log \left (c^{2} + x^{2}\right ) + 2 \, b^{2} x^{2} \log \left (x\right ) - a^{2} c^{2} + 2 \, a b c x -{\left (b^{2} c^{2} + b^{2} x^{2}\right )} \arctan \left (\frac{c}{x}\right )^{2} - 2 \,{\left (a b c^{2} - b^{2} c x\right )} \arctan \left (\frac{c}{x}\right )}{2 \, c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.43797, size = 117, normalized size = 1.39 \begin{align*} \begin{cases} - \frac{a^{2}}{2 x^{2}} - \frac{a b \operatorname{atan}{\left (\frac{c}{x} \right )}}{x^{2}} + \frac{a b}{c x} - \frac{a b \operatorname{atan}{\left (\frac{c}{x} \right )}}{c^{2}} - \frac{b^{2} \operatorname{atan}^{2}{\left (\frac{c}{x} \right )}}{2 x^{2}} + \frac{b^{2} \operatorname{atan}{\left (\frac{c}{x} \right )}}{c x} + \frac{b^{2} \log{\left (x \right )}}{c^{2}} - \frac{b^{2} \log{\left (c^{2} + x^{2} \right )}}{2 c^{2}} - \frac{b^{2} \operatorname{atan}^{2}{\left (\frac{c}{x} \right )}}{2 c^{2}} & \text{for}\: c \neq 0 \\- \frac{a^{2}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (\frac{c}{x}\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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