Optimal. Leaf size=87 \[ \frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 c^2}-\frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 x^2}-\frac{a b}{c x}-\frac{b^2 \log \left (1-\frac{c^2}{x^2}\right )}{2 c^2}-\frac{b^2 \coth ^{-1}\left (\frac{x}{c}\right )}{c x} \]
[Out]
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Rubi [C] time = 1.24698, antiderivative size = 707, normalized size of antiderivative = 8.13, 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 = {6099, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2395, 43, 6742, 30, 2557, 12, 2466, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ \frac{b^2 \text{PolyLog}\left (2,\frac{c-x}{2 c}\right )}{4 c^2}+\frac{b^2 \text{PolyLog}\left (2,-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{PolyLog}\left (2,\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{PolyLog}\left (2,\frac{c+x}{2 c}\right )}{4 c^2}-\frac{b^2 \text{PolyLog}\left (2,1-\frac{x}{c}\right )}{4 c^2}-\frac{b^2 \text{PolyLog}\left (2,\frac{x}{c}+1\right )}{4 c^2}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}-\frac{3 a b}{2 c x}+\frac{a b}{4 x^2}-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (\frac{c}{x}+1\right )^2}{16 c^2}-\frac{b^2 \left (\frac{c}{x}+1\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{b^2 \left (\frac{c}{x}+1\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 c^2}-\frac{b^2 \log \left (\frac{c}{x}+1\right ) \log (c-x)}{4 c^2}-\frac{b^2 \log (c-x) \log \left (\frac{x}{c}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac{b^2 \log \left (-\frac{x}{c}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )}{4 c^2}+\frac{b^2 \left (\frac{c}{x}+1\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^2 \left (\frac{c}{x}+1\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (\frac{c}{x}+1\right )}{4 x^2}-\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{b^2}{8 x^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6099
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 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^2}{x^3} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 x^3}+\frac{b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log \left (1+\frac{c}{x}\right )}{2 x^3}+\frac{b^2 \log ^2\left (1+\frac{c}{x}\right )}{4 x^3}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{x^3} \, dx+\frac{1}{2} b \int \frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{1}{4} b^2 \int \frac{\log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int x (2 a-b \log (1-c x))^2 \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} b \int \left (\frac{2 a \log \left (1+\frac{c}{x}\right )}{x^3}-\frac{b \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3}\right ) \, dx-\frac{1}{4} b^2 \operatorname{Subst}\left (\int x \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x))^2}{c}-\frac{(1-c x) (2 a-b \log (1-c x))^2}{c}\right ) \, dx,x,\frac{1}{x}\right )\right )+(a b) \int \frac{\log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{4} b^2 \operatorname{Subst}\left (\int \left (-\frac{\log ^2(1+c x)}{c}+\frac{(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{2} b^2 \int \frac{\log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx\\ &=\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-(a b) \operatorname{Subst}\left (\int x \log (1+c x) \, dx,x,\frac{1}{x}\right )+\frac{1}{2} b^2 \int \frac{c \log \left (1-\frac{c}{x}\right )}{2 x^3 (c+x)} \, dx+\frac{1}{2} b^2 \int \frac{c \log \left (1+\frac{c}{x}\right )}{(2 c-2 x) x^3} \, dx-\frac{\operatorname{Subst}\left (\int (2 a-b \log (1-c x))^2 \, dx,x,\frac{1}{x}\right )}{4 c}+\frac{\operatorname{Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^2 \, dx,x,\frac{1}{x}\right )}{4 c}+\frac{b^2 \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{b^2 \operatorname{Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}+\frac{\operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-\frac{c}{x}\right )}{4 c^2}-\frac{\operatorname{Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}+\frac{1}{2} (a b c) \operatorname{Subst}\left (\int \frac{x^2}{1+c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (b^2 c\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{x^3 (c+x)} \, dx+\frac{1}{2} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{(2 c-2 x) x^3} \, dx\\ &=\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b \operatorname{Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-\frac{c}{x}\right )}{4 c^2}+\frac{b \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-\frac{c}{x}\right )}{2 c^2}+\frac{b^2 \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{c}{x}\right )}{2 c^2}+\frac{1}{2} (a b c) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}+\frac{x}{c}+\frac{1}{c^2 (1+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (b^2 c\right ) \int \left (\frac{\log \left (1-\frac{c}{x}\right )}{c x^3}-\frac{\log \left (1-\frac{c}{x}\right )}{c^2 x^2}+\frac{\log \left (1-\frac{c}{x}\right )}{c^3 x}-\frac{\log \left (1-\frac{c}{x}\right )}{c^3 (c+x)}\right ) \, dx+\frac{1}{2} \left (b^2 c\right ) \int \left (\frac{\log \left (1+\frac{c}{x}\right )}{2 c^3 (c-x)}+\frac{\log \left (1+\frac{c}{x}\right )}{2 c x^3}+\frac{\log \left (1+\frac{c}{x}\right )}{2 c^2 x^2}+\frac{\log \left (1+\frac{c}{x}\right )}{2 c^3 x}\right ) \, dx\\ &=-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{a b}{4 x^2}-\frac{3 a b}{2 c x}+\frac{b^2}{2 c x}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{2 c^2}+\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{1}{4} b^2 \int \frac{\log \left (1-\frac{c}{x}\right )}{x^3} \, dx+\frac{1}{4} b^2 \int \frac{\log \left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{b^2 \int \frac{\log \left (1-\frac{c}{x}\right )}{x} \, dx}{4 c^2}-\frac{b^2 \int \frac{\log \left (1-\frac{c}{x}\right )}{c+x} \, dx}{4 c^2}+\frac{b^2 \int \frac{\log \left (1+\frac{c}{x}\right )}{c-x} \, dx}{4 c^2}+\frac{b^2 \int \frac{\log \left (1+\frac{c}{x}\right )}{x} \, dx}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{c}{x}\right )}{2 c^2}-\frac{b^2 \int \frac{\log \left (1-\frac{c}{x}\right )}{x^2} \, dx}{4 c}+\frac{b^2 \int \frac{\log \left (1+\frac{c}{x}\right )}{x^2} \, dx}{4 c}\\ &=-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{a b}{4 x^2}-\frac{3 a b}{2 c x}-\frac{b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{2 c^2}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{2 c^2}+\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}-\frac{1}{4} b^2 \operatorname{Subst}\left (\int x \log (1-c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{4} b^2 \operatorname{Subst}\left (\int x \log (1+c x) \, dx,x,\frac{1}{x}\right )-\frac{b^2 \int \frac{\log (c-x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx}{4 c}+\frac{b^2 \int \frac{\log (c+x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx}{4 c}+\frac{b^2 \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{b^2 \operatorname{Subst}\left (\int \log (1+c x) \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{a b}{4 x^2}-\frac{3 a b}{2 c x}-\frac{b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{2 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{2 c^2}+\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}-\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \int \left (\frac{\log (c-x)}{c x}-\frac{\log (c-x)}{c (c+x)}\right ) \, dx}{4 c}+\frac{b^2 \int \left (-\frac{\log (c+x)}{c (c-x)}-\frac{\log (c+x)}{c x}\right ) \, dx}{4 c}-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{a b}{4 x^2}-\frac{3 a b}{2 c x}-\frac{3 b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}-\frac{3 b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}-\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \int \frac{\log (c-x)}{x} \, dx}{4 c^2}+\frac{b^2 \int \frac{\log (c-x)}{c+x} \, dx}{4 c^2}-\frac{b^2 \int \frac{\log (c+x)}{c-x} \, dx}{4 c^2}-\frac{b^2 \int \frac{\log (c+x)}{x} \, dx}{4 c^2}-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}+\frac{x}{c}+\frac{1}{c^2 (1+c x)}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{a b}{4 x^2}+\frac{b^2}{8 x^2}-\frac{3 a b}{2 c x}+\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{b^2 \log (c-x) \log \left (\frac{x}{c}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac{b^2 \log \left (-\frac{x}{c}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )}{4 c^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}+\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}-\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \int \frac{\log \left (-\frac{-c-x}{2 c}\right )}{c-x} \, dx}{4 c^2}-\frac{b^2 \int \frac{\log \left (\frac{c-x}{2 c}\right )}{c+x} \, dx}{4 c^2}+\frac{b^2 \int \frac{\log \left (-\frac{x}{c}\right )}{c+x} \, dx}{4 c^2}-\frac{b^2 \int \frac{\log \left (\frac{x}{c}\right )}{c-x} \, dx}{4 c^2}\\ &=-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{a b}{4 x^2}+\frac{b^2}{8 x^2}-\frac{3 a b}{2 c x}+\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{b^2 \log (c-x) \log \left (\frac{x}{c}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac{b^2 \log \left (-\frac{x}{c}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )}{4 c^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}+\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}-\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1-\frac{x}{c}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1+\frac{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-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+x\right )}{4 c^2}\\ &=-\frac{b^2 \left (1-\frac{c}{x}\right )^2}{16 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{a b}{4 x^2}+\frac{b^2}{8 x^2}-\frac{3 a b}{2 c x}+\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}+\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{b^2 \log (c-x) \log \left (\frac{x}{c}\right )}{4 c^2}-\frac{b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac{b^2 \log \left (-\frac{x}{c}\right ) \log (c+x)}{4 c^2}+\frac{b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )}{4 c^2}+\frac{a b \log \left (\frac{c+x}{x}\right )}{2 c^2}+\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{a b \log \left (\frac{c+x}{x}\right )}{2 x^2}-\frac{b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c-x}{2 c}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}+\frac{b^2 \text{Li}_2\left (\frac{c+x}{2 c}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1-\frac{x}{c}\right )}{4 c^2}-\frac{b^2 \text{Li}_2\left (1+\frac{x}{c}\right )}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.067157, size = 119, normalized size = 1.37 \[ -\frac{a^2 c^2+a b x^2 \log (x-c)-a b x^2 \log (c+x)+2 a b c x+2 b c \tanh ^{-1}\left (\frac{c}{x}\right ) (a c+b x)+b^2 \left (c^2-x^2\right ) \tanh ^{-1}\left (\frac{c}{x}\right )^2+b^2 x^2 \log (x-c)+b^2 x^2 \log (c+x)-2 b^2 x^2 \log (x)}{2 c^2 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 284, normalized size = 3.3 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{{b}^{2}}{2\,{x}^{2}} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{{b}^{2}}{cx}{\it Artanh} \left ({\frac{c}{x}} \right ) }-{\frac{{b}^{2}}{2\,{c}^{2}}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{{b}^{2}}{2\,{c}^{2}}{\it Artanh} \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{{b}^{2}}{4\,{c}^{2}}\ln \left ({\frac{c}{x}}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }-{\frac{{b}^{2}}{8\,{c}^{2}} \left ( \ln \left ({\frac{c}{x}}-1 \right ) \right ) ^{2}}-{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ({\frac{c}{x}}-1 \right ) }-{\frac{{b}^{2}}{2\,{c}^{2}}\ln \left ( 1+{\frac{c}{x}} \right ) }+{\frac{{b}^{2}}{4\,{c}^{2}}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) }-{\frac{{b}^{2}}{4\,{c}^{2}}\ln \left ( -{\frac{c}{2\,x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2\,x}} \right ) }-{\frac{{b}^{2}}{8\,{c}^{2}} \left ( \ln \left ( 1+{\frac{c}{x}} \right ) \right ) ^{2}}-{\frac{ab}{{x}^{2}}{\it Artanh} \left ({\frac{c}{x}} \right ) }-{\frac{ab}{cx}}-{\frac{ab}{2\,{c}^{2}}\ln \left ({\frac{c}{x}}-1 \right ) }+{\frac{ab}{2\,{c}^{2}}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.989422, size = 223, normalized size = 2.56 \begin{align*} \frac{1}{2} \,{\left (c{\left (\frac{\log \left (c + x\right )}{c^{3}} - \frac{\log \left (-c + x\right )}{c^{3}} - \frac{2}{c^{2} x}\right )} - \frac{2 \, \operatorname{artanh}\left (\frac{c}{x}\right )}{x^{2}}\right )} a b - \frac{1}{8} \,{\left (c^{2}{\left (\frac{\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 )}{c^{4}} - \frac{8 \, \log \left (x\right )}{c^{4}}\right )} - 4 \, c{\left (\frac{\log \left (c + x\right )}{c^{3}} - \frac{\log \left (-c + x\right )}{c^{3}} - \frac{2}{c^{2} x}\right )} \operatorname{artanh}\left (\frac{c}{x}\right )\right )} b^{2} - \frac{b^{2} \operatorname{artanh}\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 = 1.86745, size = 288, normalized size = 3.31 \begin{align*} \frac{8 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a^{2} c^{2} - 8 \, a b c x + 4 \,{\left (a b - b^{2}\right )} x^{2} \log \left (c + x\right ) - 4 \,{\left (a b + b^{2}\right )} x^{2} \log \left (-c + x\right ) -{\left (b^{2} c^{2} - b^{2} x^{2}\right )} \log \left (-\frac{c + x}{c - x}\right )^{2} - 4 \,{\left (a b c^{2} + b^{2} c x\right )} \log \left (-\frac{c + x}{c - x}\right )}{8 \, c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.96205, size = 124, normalized size = 1.43 \begin{align*} \begin{cases} - \frac{a^{2}}{2 x^{2}} - \frac{a b \operatorname{atanh}{\left (\frac{c}{x} \right )}}{x^{2}} - \frac{a b}{c x} + \frac{a b \operatorname{atanh}{\left (\frac{c}{x} \right )}}{c^{2}} - \frac{b^{2} \operatorname{atanh}^{2}{\left (\frac{c}{x} \right )}}{2 x^{2}} - \frac{b^{2} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{c x} + \frac{b^{2} \log{\left (x \right )}}{c^{2}} - \frac{b^{2} \log{\left (- c + x \right )}}{c^{2}} + \frac{b^{2} \operatorname{atanh}^{2}{\left (\frac{c}{x} \right )}}{2 c^{2}} - \frac{b^{2} \operatorname{atanh}{\left (\frac{c}{x} \right )}}{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 \operatorname{artanh}\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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