Optimal. Leaf size=94 \[ -\frac{1}{4} c^2 \left (a+b \coth ^{-1}\left (\frac{x^2}{c}\right )\right )^2+\frac{1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac{x^2}{c}\right )\right )^2+\frac{1}{2} b c x^2 \left (a+b \coth ^{-1}\left (\frac{x^2}{c}\right )\right )+\frac{1}{4} b^2 c^2 \log \left (1-\frac{c^2}{x^4}\right )+b^2 c^2 \log (x) \]
[Out]
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Rubi [C] time = 1.29574, antiderivative size = 599, normalized size of antiderivative = 6.37, number of steps used = 59, number of rules used = 33, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.063, Rules used = {6099, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2455, 263, 266, 43, 193, 6742, 30, 2557, 12, 2466, 2448, 2462, 260, 2416, 2394, 2393, 2391, 2410, 2395, 36, 29, 2390} \[ -\frac{1}{8} b^2 c^2 \text{PolyLog}\left (2,-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{PolyLog}\left (2,\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{PolyLog}\left (2,\frac{c-x^2}{2 c}\right )-\frac{1}{8} b^2 c^2 \text{PolyLog}\left (2,\frac{c+x^2}{2 c}\right )+\frac{1}{8} b^2 c^2 \text{PolyLog}\left (2,\frac{c+x^2}{c}\right )+\frac{1}{8} b^2 c^2 \text{PolyLog}\left (2,1-\frac{x^2}{c}\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2-\frac{1}{4} a b c^2 \log \left (c+x^2\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{4} a b c x^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{4} a b x^4 \log \left (\frac{c}{x^2}+1\right )+\frac{1}{8} b c x^2 \left (1-\frac{c}{x^2}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} b^2 c^2 \log ^2\left (\frac{c}{x^2}+1\right )+\frac{1}{8} b^2 c^2 \log \left (\frac{c}{x^2}+1\right )+\frac{1}{8} b^2 c^2 \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (\frac{c}{x^2}+1\right ) \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (\frac{x^2}{c}\right ) \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (-\frac{x^2}{c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (\frac{c-x^2}{2 c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (c-x^2\right ) \log \left (\frac{c+x^2}{2 c}\right )+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{16} b^2 x^4 \log ^2\left (\frac{c}{x^2}+1\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (\frac{c}{x^2}+1\right )-\frac{1}{8} b^2 c x^2 \log \left (1-\frac{c}{x^2}\right )+\frac{1}{4} b^2 c x^2 \log \left (\frac{c}{x^2}+1\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6099
Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2316
Rule 2315
Rule 2314
Rule 31
Rule 2455
Rule 263
Rule 266
Rule 43
Rule 193
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 2410
Rule 2395
Rule 36
Rule 29
Rule 2390
Rubi steps
\begin{align*} \int x^3 \left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2-\frac{1}{2} b x^3 \left (-2 a+b \log \left (1-\frac{c}{x^2}\right )\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 x^3 \log ^2\left (1+\frac{c}{x^2}\right )\right ) \, dx\\ &=\frac{1}{4} \int x^3 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2 \, dx-\frac{1}{2} b \int x^3 \left (-2 a+b \log \left (1-\frac{c}{x^2}\right )\right ) \log \left (1+\frac{c}{x^2}\right ) \, dx+\frac{1}{4} b^2 \int x^3 \log ^2\left (1+\frac{c}{x^2}\right ) \, dx\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{x^3} \, dx,x,\frac{1}{x^2}\right )\right )-\frac{1}{4} b \operatorname{Subst}\left (\int x \left (-2 a+b \log \left (1-\frac{c}{x}\right )\right ) \log \left (1+\frac{c}{x}\right ) \, dx,x,x^2\right )-\frac{1}{8} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^3} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )-\frac{1}{4} b \operatorname{Subst}\left (\int \left (-2 a x \log \left (1+\frac{c}{x}\right )+b x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )\right ) \, dx,x,x^2\right )-\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{x^2 (1-c x)} \, dx,x,\frac{1}{x^2}\right )-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2 (1+c x)} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{8} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x^2}\right )+\frac{1}{2} (a b) \operatorname{Subst}\left (\int x \log \left (1+\frac{c}{x}\right ) \, dx,x,x^2\right )-\frac{1}{4} b^2 \operatorname{Subst}\left (\int x \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right ) \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^2}-\frac{c \log (1+c x)}{x}+\frac{c^2 \log (1+c x)}{1+c x}\right ) \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{8} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x^2}\right )+\frac{1}{4} b^2 \operatorname{Subst}\left (\int \frac{c x \log \left (1-\frac{c}{x}\right )}{2 (-c-x)} \, dx,x,x^2\right )+\frac{1}{4} b^2 \operatorname{Subst}\left (\int \frac{c x \log \left (1+\frac{c}{x}\right )}{-2 c+2 x} \, dx,x,x^2\right )+\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-\frac{c}{x^2}\right )+\frac{1}{4} (a b c) \operatorname{Subst}\left (\int \frac{1}{1+\frac{c}{x}} \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2} \, dx,x,\frac{1}{x^2}\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x} \, dx,x,\frac{1}{x^2}\right )-\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )+\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x^2}\right )+\frac{1}{4} (a b c) \operatorname{Subst}\left (\int \frac{x}{c+x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x^2}\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x \log \left (1-\frac{c}{x}\right )}{-c-x} \, dx,x,x^2\right )+\frac{1}{4} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x \log \left (1+\frac{c}{x}\right )}{-2 c+2 x} \, dx,x,x^2\right )+\frac{1}{8} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-\frac{c}{x^2}\right )-\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+c x)} \, dx,x,\frac{1}{x^2}\right )-\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{c}{x^2}\right )\\ &=\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{4} b^2 c^2 \log (x)-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )+\frac{1}{4} (a b c) \operatorname{Subst}\left (\int \left (1-\frac{c}{c+x}\right ) \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (-\log \left (1-\frac{c}{x}\right )+\frac{c \log \left (1-\frac{c}{x}\right )}{c+x}\right ) \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x^2}\right )+\frac{1}{4} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2} \log \left (1+\frac{c}{x}\right )-\frac{c \log \left (1+\frac{c}{x}\right )}{2 (c-x)}\right ) \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x^2}\right )+\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c x} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{4} a b c x^2+\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{8} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{4} a b c^2 \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c}{x^2}\right )-\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{c}{x}\right ) \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{c}{x}\right ) \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{c}{x}\right )}{c+x} \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{c}{x}\right )}{c-x} \, dx,x,x^2\right )\\ &=\frac{1}{4} a b c x^2-\frac{1}{8} b^2 c x^2 \log \left (1-\frac{c}{x^2}\right )+\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right ) \log \left (c-x^2\right )-\frac{1}{4} a b c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c}{x^2}\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{c}{x}\right ) x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{c}{x}\right ) x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (c-x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (c+x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx,x,x^2\right )\\ &=\frac{1}{4} a b c x^2-\frac{1}{8} b^2 c x^2 \log \left (1-\frac{c}{x^2}\right )+\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right ) \log \left (c-x^2\right )-\frac{1}{4} a b c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c}{x^2}\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-c+x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{\log (c-x)}{c x}-\frac{\log (c-x)}{c (c+x)}\right ) \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (c+x)}{c (c-x)}-\frac{\log (c+x)}{c x}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{4} a b c x^2-\frac{1}{8} b^2 c x^2 \log \left (1-\frac{c}{x^2}\right )+\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{8} b^2 c^2 \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right ) \log \left (c-x^2\right )-\frac{1}{4} a b c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c}{x^2}\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (c-x)}{x} \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (c-x)}{c+x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (c+x)}{c-x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (c+x)}{x} \, dx,x,x^2\right )\\ &=\frac{1}{4} a b c x^2-\frac{1}{8} b^2 c x^2 \log \left (1-\frac{c}{x^2}\right )+\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{8} b^2 c^2 \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right ) \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (\frac{x^2}{c}\right ) \log \left (c-x^2\right )-\frac{1}{4} a b c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (-\frac{x^2}{c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (\frac{c-x^2}{2 c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (c-x^2\right ) \log \left (\frac{c+x^2}{2 c}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c}{x^2}\right )-\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-c-x}{2 c}\right )}{c-x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{c-x}{2 c}\right )}{c+x} \, dx,x,x^2\right )-\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{x}{c}\right )}{c+x} \, dx,x,x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{x}{c}\right )}{c-x} \, dx,x,x^2\right )\\ &=\frac{1}{4} a b c x^2-\frac{1}{8} b^2 c x^2 \log \left (1-\frac{c}{x^2}\right )+\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{8} b^2 c^2 \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right ) \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (\frac{x^2}{c}\right ) \log \left (c-x^2\right )-\frac{1}{4} a b c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (-\frac{x^2}{c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (\frac{c-x^2}{2 c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (c-x^2\right ) \log \left (\frac{c+x^2}{2 c}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c}{x^2}\right )+\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c+x^2}{c}\right )+\frac{1}{8} b^2 c^2 \text{Li}_2\left (1-\frac{x^2}{c}\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-x^2\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+x^2\right )\\ &=\frac{1}{4} a b c x^2-\frac{1}{8} b^2 c x^2 \log \left (1-\frac{c}{x^2}\right )+\frac{1}{8} b c \left (1-\frac{c}{x^2}\right ) x^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )-\frac{1}{16} c^2 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{16} x^4 \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} b^2 c x^2 \log \left (1+\frac{c}{x^2}\right )+\frac{1}{4} a b x^4 \log \left (1+\frac{c}{x^2}\right )-\frac{1}{8} b^2 x^4 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )-\frac{1}{16} b^2 c^2 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{16} b^2 x^4 \log ^2\left (1+\frac{c}{x^2}\right )+\frac{1}{2} a b c^2 \log (x)+\frac{1}{2} b^2 c^2 \log (x)+\frac{1}{8} b^2 c^2 \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1+\frac{c}{x^2}\right ) \log \left (c-x^2\right )+\frac{1}{8} b^2 c^2 \log \left (\frac{x^2}{c}\right ) \log \left (c-x^2\right )-\frac{1}{4} a b c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (c+x^2\right )+\frac{1}{8} b^2 c^2 \log \left (-\frac{x^2}{c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (\frac{c-x^2}{2 c}\right ) \log \left (c+x^2\right )-\frac{1}{8} b^2 c^2 \log \left (c-x^2\right ) \log \left (\frac{c+x^2}{2 c}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (-\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c}{x^2}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c-x^2}{2 c}\right )-\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c+x^2}{2 c}\right )+\frac{1}{8} b^2 c^2 \text{Li}_2\left (\frac{c+x^2}{c}\right )+\frac{1}{8} b^2 c^2 \text{Li}_2\left (1-\frac{x^2}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.0604428, size = 104, normalized size = 1.11 \[ \frac{1}{4} \left (a^2 x^4+b c^2 (a+b) \log \left (x^2-c\right )-a b c^2 \log \left (c+x^2\right )+2 a b c x^2+2 b x^2 \tanh ^{-1}\left (\frac{c}{x^2}\right ) \left (a x^2+b c\right )+b^2 c^2 \log \left (c+x^2\right )+b^2 \left (x^4-c^2\right ) \tanh ^{-1}\left (\frac{c}{x^2}\right )^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978379, size = 212, normalized size = 2.26 \begin{align*} \frac{1}{4} \, b^{2} x^{4} \operatorname{artanh}\left (\frac{c}{x^{2}}\right )^{2} + \frac{1}{4} \, a^{2} x^{4} + \frac{1}{4} \,{\left (2 \, x^{4} \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) +{\left (2 \, x^{2} - c \log \left (x^{2} + c\right ) + c \log \left (x^{2} - c\right )\right )} c\right )} a b + \frac{1}{16} \,{\left ({\left (\log \left (x^{2} + c\right )^{2} - 2 \,{\left (\log \left (x^{2} + c\right ) - 2\right )} \log \left (x^{2} - c\right ) + \log \left (x^{2} - c\right )^{2} + 4 \, \log \left (x^{2} + c\right )\right )} c^{2} + 4 \,{\left (2 \, x^{2} - c \log \left (x^{2} + c\right ) + c \log \left (x^{2} - c\right )\right )} c \operatorname{artanh}\left (\frac{c}{x^{2}}\right )\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81725, size = 278, normalized size = 2.96 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + \frac{1}{2} \, a b c x^{2} - \frac{1}{4} \,{\left (a b - b^{2}\right )} c^{2} \log \left (x^{2} + c\right ) + \frac{1}{4} \,{\left (a b + b^{2}\right )} c^{2} \log \left (x^{2} - c\right ) + \frac{1}{16} \,{\left (b^{2} x^{4} - b^{2} c^{2}\right )} \log \left (\frac{x^{2} + c}{x^{2} - c}\right )^{2} + \frac{1}{4} \,{\left (a b x^{4} + b^{2} c x^{2}\right )} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 14.7646, size = 151, normalized size = 1.61 \begin{align*} \frac{a^{2} x^{4}}{4} - \frac{a b c^{2} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} + \frac{a b c x^{2}}{2} + \frac{a b x^{4} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} + \frac{b^{2} c^{2} \log{\left (- i \sqrt{c} + x \right )}}{2} + \frac{b^{2} c^{2} \log{\left (i \sqrt{c} + x \right )}}{2} - \frac{b^{2} c^{2} \operatorname{atanh}^{2}{\left (\frac{c}{x^{2}} \right )}}{4} - \frac{b^{2} c^{2} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} + \frac{b^{2} c x^{2} \operatorname{atanh}{\left (\frac{c}{x^{2}} \right )}}{2} + \frac{b^{2} x^{4} \operatorname{atanh}^{2}{\left (\frac{c}{x^{2}} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2751, size = 180, normalized size = 1.91 \begin{align*} \frac{1}{4} \, a^{2} x^{4} + \frac{1}{2} \, a b c x^{2} + \frac{1}{16} \,{\left (b^{2} x^{4} - b^{2} c^{2}\right )} \log \left (\frac{x^{2} + c}{x^{2} - c}\right )^{2} - \frac{1}{4} \,{\left (a b c^{2} - b^{2} c^{2}\right )} \log \left (x^{2} + c\right ) + \frac{1}{4} \,{\left (a b c^{2} + b^{2} c^{2}\right )} \log \left (-x^{2} + c\right ) + \frac{1}{4} \,{\left (a b x^{4} + b^{2} c x^{2}\right )} \log \left (\frac{x^{2} + c}{x^{2} - c}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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