Optimal. Leaf size=144 \[ -\frac{1}{9} b^2 c^3 \text{PolyLog}\left (2,\frac{2}{c x^3+1}-1\right )+\frac{1}{9} c^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac{2}{9} b c^3 \log \left (2-\frac{2}{c x^3+1}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-\frac{b c \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{9 x^6}-\frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{9 x^9}-\frac{b^2 c^2}{9 x^3}+\frac{1}{9} b^2 c^3 \tanh ^{-1}\left (c x^3\right ) \]
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Rubi [B] time = 1.31132, antiderivative size = 420, normalized size of antiderivative = 2.92, number of steps used = 59, number of rules used = 24, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.5, Rules used = {6099, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 2395, 2439, 2416, 36, 29, 2392, 2391, 2394, 2393, 2410, 2390} \[ -\frac{1}{9} b^2 c^3 \text{PolyLog}\left (2,-c x^3\right )+\frac{1}{9} b^2 c^3 \text{PolyLog}\left (2,c x^3\right )+\frac{1}{18} b^2 c^3 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{18} b^2 c^3 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^3+1\right )\right )+\frac{1}{36} c^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{1}{18} b c^3 \log \left (\frac{1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{2}{3} a b c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^9}-\frac{b^2 c^2}{9 x^3}-\frac{1}{36} b^2 c^3 \log ^2\left (c x^3+1\right )+\frac{1}{18} b^2 c^3 \log \left (c x^3+1\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )-\frac{b^2 \log ^2\left (c x^3+1\right )}{36 x^9}-\frac{b^2 c \log \left (c x^3+1\right )}{18 x^6} \]
Warning: Unable to verify antiderivative.
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Rule 6099
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 2395
Rule 2439
Rule 2416
Rule 36
Rule 29
Rule 2392
Rule 2391
Rule 2394
Rule 2393
Rule 2410
Rule 2390
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{x^{10}} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{4 x^{10}}-\frac{b \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{2 x^{10}}+\frac{b^2 \log ^2\left (1+c x^3\right )}{4 x^{10}}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{x^{10}} \, dx-\frac{1}{2} b \int \frac{\left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{x^{10}} \, dx+\frac{1}{4} b^2 \int \frac{\log ^2\left (1+c x^3\right )}{x^{10}} \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{x^4} \, dx,x,x^3\right )-\frac{1}{6} b \operatorname{Subst}\left (\int \frac{(-2 a+b \log (1-c x)) \log (1+c x)}{x^4} \, dx,x,x^3\right )+\frac{1}{12} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^4} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}+\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{x^3 (1-c x)} \, dx,x,x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^3 (1+c x)} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3 (1-c x)} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3 (1+c x)} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{18} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \left (\frac{-2 a+b \log (1-c x)}{x^3}-\frac{c (-2 a+b \log (1-c x))}{x^2}+\frac{c^2 (-2 a+b \log (1-c x))}{x}-\frac{c^3 (-2 a+b \log (1-c x))}{1+c x}\right ) \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^3}+\frac{c \log (1+c x)}{x^2}+\frac{c^2 \log (1+c x)}{x}-\frac{c^3 \log (1+c x)}{-1+c x}\right ) \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^3}-\frac{c \log (1+c x)}{x^2}+\frac{c^2 \log (1+c x)}{x}-\frac{c^3 \log (1+c x)}{1+c x}\right ) \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{18} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^3} \, dx,x,x^3\right )+2 \left (\frac{1}{18} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3} \, dx,x,x^3\right )\right )+\frac{1}{18} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{18} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x} \, dx,x,x^3\right )+2 \left (\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x} \, dx,x,x^3\right )\right )+\frac{1}{18} \left (b c^4\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=\frac{1}{3} a b c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )-\frac{1}{18} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{36} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{18} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-c x^3\right )+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1-c x)} \, dx,x,x^3\right )+2 \left (-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+c x)} \, dx,x,x^3\right )\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x (1-c x)} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+c x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )+\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )\\ &=\frac{1}{3} a b c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{1}{36} b^2 c^3 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (c x^3\right )-\frac{1}{36} \left (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-c x^3\right )-\frac{1}{18} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c}{x}-\frac{c^2}{-1+c x}\right ) \, dx,x,x^3\right )+2 \left (-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}+\frac{1}{36} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c}{x}+\frac{c^2}{1+c x}\right ) \, dx,x,x^3\right )\right )-\frac{1}{18} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )-\frac{1}{18} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-c x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )-\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )+\frac{1}{18} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )-\frac{1}{18} \left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x} \, dx,x,x^3\right )\\ &=-\frac{b^2 c^2}{18 x^3}+\frac{2}{3} a b c^3 \log (x)+\frac{1}{6} b^2 c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{1}{36} c^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{1}{36} b^2 c^3 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}+2 \left (-\frac{b^2 c^2}{36 x^3}-\frac{1}{12} b^2 c^3 \log (x)+\frac{1}{36} b^2 c^3 \log \left (1+c x^3\right )-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}\right )-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )+\frac{1}{18} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )\\ &=-\frac{b^2 c^2}{18 x^3}+\frac{2}{3} a b c^3 \log (x)+\frac{1}{6} b^2 c^3 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^6}+\frac{b c^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}-\frac{b c^2 \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{18 x^3}+\frac{1}{36} c^3 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{36 x^9}-\frac{1}{18} b c^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{18} b^2 c^3 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{18 x^9}-\frac{1}{36} b^2 c^3 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{36 x^9}+2 \left (-\frac{b^2 c^2}{36 x^3}-\frac{1}{12} b^2 c^3 \log (x)+\frac{1}{36} b^2 c^3 \log \left (1+c x^3\right )-\frac{b^2 c \log \left (1+c x^3\right )}{36 x^6}\right )-\frac{1}{9} b^2 c^3 \text{Li}_2\left (-c x^3\right )+\frac{1}{9} b^2 c^3 \text{Li}_2\left (c x^3\right )+\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{18} b^2 c^3 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )\\ \end{align*}
Mathematica [A] time = 0.362807, size = 159, normalized size = 1.1 \[ -\frac{b^2 c^3 x^9 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+a^2-2 a b c^3 x^9 \log \left (c x^3\right )+a b c^3 x^9 \log \left (1-c^2 x^6\right )+b \tanh ^{-1}\left (c x^3\right ) \left (2 a-b c^3 x^9-2 b c^3 x^9 \log \left (1-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )+b c x^3\right )+a b c x^3+b^2 c^2 x^6+b^2 \left (1-c^3 x^9\right ) \tanh ^{-1}\left (c x^3\right )^2}{9 x^9} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( c{x}^{3} \right ) \right ) ^{2}}{{x}^{10}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{9} \,{\left ({\left (c^{2} \log \left (c^{2} x^{6} - 1\right ) - c^{2} \log \left (x^{6}\right ) + \frac{1}{x^{6}}\right )} c + \frac{2 \, \operatorname{artanh}\left (c x^{3}\right )}{x^{9}}\right )} a b - \frac{1}{36} \, b^{2}{\left (\frac{\log \left (-c x^{3} + 1\right )^{2}}{x^{9}} + 9 \, \int -\frac{3 \,{\left (c x^{3} - 1\right )} \log \left (c x^{3} + 1\right )^{2} + 2 \,{\left (c x^{3} - 3 \,{\left (c x^{3} - 1\right )} \log \left (c x^{3} + 1\right )\right )} \log \left (-c x^{3} + 1\right )}{3 \,{\left (c x^{13} - x^{10}\right )}}\,{d x}\right )} - \frac{a^{2}}{9 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (c x^{3}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{3}\right ) + a^{2}}{x^{10}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (c x^{3}\right ) + a\right )}^{2}}{x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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