Optimal. Leaf size=415 \[ -\frac{26 b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{35 c^4}+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac{3 a b d^3 x}{2 c^3}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}-\frac{52 b d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{35 c^4}+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{7 b^2 d^3 x^2}{20 c^2}+\frac{11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}+\frac{122 b^2 d^3 x}{105 c^3}+\frac{3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}-\frac{122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac{1}{105} b^2 c d^3 x^5+\frac{44 b^2 d^3 x^3}{315 c}+\frac{1}{20} b^2 d^3 x^4 \]
[Out]
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Rubi [A] time = 1.45901, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 62, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {5940, 5916, 5980, 266, 43, 5910, 260, 5948, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac{26 b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{35 c^4}+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac{3 a b d^3 x}{2 c^3}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}-\frac{52 b d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{35 c^4}+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{7 b^2 d^3 x^2}{20 c^2}+\frac{11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}+\frac{122 b^2 d^3 x}{105 c^3}+\frac{3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}-\frac{122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac{1}{105} b^2 c d^3 x^5+\frac{44 b^2 d^3 x^3}{315 c}+\frac{1}{20} b^2 d^3 x^4 \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5916
Rule 5980
Rule 266
Rule 43
Rule 5910
Rule 260
Rule 5948
Rule 302
Rule 206
Rule 321
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x^3 (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+3 c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c d^3\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (3 c^2 d^3\right ) \int x^5 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^3 d^3\right ) \int x^6 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{2} \left (b c d^3\right ) \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{5} \left (6 b c^2 d^3\right ) \int \frac{x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b c^3 d^3\right ) \int \frac{x^6 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{7} \left (2 b c^4 d^3\right ) \int \frac{x^7 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} \left (6 b d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{5} \left (6 b d^3\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac{\left (b d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}-\frac{\left (b d^3\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{2 c}+\left (b c d^3\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (b c d^3\right ) \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac{1}{7} \left (2 b c^2 d^3\right ) \int x^5 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{7} \left (2 b c^2 d^3\right ) \int \frac{x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac{3}{10} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} \left (2 b d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{7} \left (2 b d^3\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{6} \left (b^2 d^3\right ) \int \frac{x^3}{1-c^2 x^2} \, dx+\frac{\left (b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac{\left (b d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^3}+\frac{\left (6 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac{\left (6 b d^3\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c^2}+\frac{\left (b d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac{\left (b d^3\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c}-\frac{1}{10} \left (3 b^2 c d^3\right ) \int \frac{x^4}{1-c^2 x^2} \, dx-\frac{1}{5} \left (b^2 c^2 d^3\right ) \int \frac{x^5}{1-c^2 x^2} \, dx-\frac{1}{21} \left (b^2 c^3 d^3\right ) \int \frac{x^6}{1-c^2 x^2} \, dx\\ &=\frac{a b d^3 x}{2 c^3}+\frac{3 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{7 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{12} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{3} \left (b^2 d^3\right ) \int \frac{x^3}{1-c^2 x^2} \, dx+\frac{\left (b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^3}-\frac{\left (b d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^3}-\frac{\left (6 b d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^3}+\frac{\left (b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^3}+\frac{\left (2 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{7 c^2}-\frac{\left (2 b d^3\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{7 c^2}-\frac{\left (3 b^2 d^3\right ) \int \frac{x^2}{1-c^2 x^2} \, dx}{5 c}-\frac{1}{14} \left (b^2 c d^3\right ) \int \frac{x^4}{1-c^2 x^2} \, dx-\frac{1}{10} \left (3 b^2 c d^3\right ) \int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx-\frac{1}{10} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{21} \left (b^2 c^3 d^3\right ) \int \left (-\frac{1}{c^6}-\frac{x^2}{c^4}-\frac{x^4}{c^2}+\frac{1}{c^6 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{3 a b d^3 x}{2 c^3}+\frac{199 b^2 d^3 x}{210 c^3}+\frac{73 b^2 d^3 x^3}{630 c}+\frac{1}{105} b^2 c d^3 x^5+\frac{b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac{26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{6 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{5 c^4}-\frac{1}{12} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{6} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{\left (2 b d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{7 c^3}-\frac{\left (b^2 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx}{21 c^3}-\frac{\left (3 b^2 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx}{10 c^3}-\frac{\left (3 b^2 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx}{5 c^3}+\frac{\left (b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{c^3}+\frac{\left (6 b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^3}-\frac{\left (b^2 d^3\right ) \int \frac{x}{1-c^2 x^2} \, dx}{2 c^2}-\frac{\left (b^2 d^3\right ) \int \frac{x^2}{1-c^2 x^2} \, dx}{7 c}-\frac{1}{14} \left (b^2 c d^3\right ) \int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx-\frac{1}{10} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^2}-\frac{1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a b d^3 x}{2 c^3}+\frac{122 b^2 d^3 x}{105 c^3}+\frac{11 b^2 d^3 x^2}{60 c^2}+\frac{44 b^2 d^3 x^3}{315 c}+\frac{1}{20} b^2 d^3 x^4+\frac{1}{105} b^2 c d^3 x^5-\frac{199 b^2 d^3 \tanh ^{-1}(c x)}{210 c^4}+\frac{3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac{26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{52 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{35 c^4}+\frac{13 b^2 d^3 \log \left (1-c^2 x^2\right )}{30 c^4}-\frac{1}{6} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (6 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{5 c^4}-\frac{\left (b^2 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx}{14 c^3}-\frac{\left (b^2 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx}{7 c^3}+\frac{\left (2 b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{7 c^3}-\frac{\left (b^2 d^3\right ) \int \frac{x}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac{3 a b d^3 x}{2 c^3}+\frac{122 b^2 d^3 x}{105 c^3}+\frac{7 b^2 d^3 x^2}{20 c^2}+\frac{44 b^2 d^3 x^3}{315 c}+\frac{1}{20} b^2 d^3 x^4+\frac{1}{105} b^2 c d^3 x^5-\frac{122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac{3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac{26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{52 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{35 c^4}+\frac{11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}-\frac{3 b^2 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{5 c^4}-\frac{\left (2 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{7 c^4}\\ &=\frac{3 a b d^3 x}{2 c^3}+\frac{122 b^2 d^3 x}{105 c^3}+\frac{7 b^2 d^3 x^2}{20 c^2}+\frac{44 b^2 d^3 x^3}{315 c}+\frac{1}{20} b^2 d^3 x^4+\frac{1}{105} b^2 c d^3 x^5-\frac{122 b^2 d^3 \tanh ^{-1}(c x)}{105 c^4}+\frac{3 b^2 d^3 x \tanh ^{-1}(c x)}{2 c^3}+\frac{26 b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{35 c^2}+\frac{b d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac{13}{35} b d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} b c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{21} b c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{140 c^4}+\frac{1}{4} d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{3}{5} c d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^2 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{7} c^3 d^3 x^7 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{52 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{35 c^4}+\frac{11 b^2 d^3 \log \left (1-c^2 x^2\right )}{10 c^4}-\frac{26 b^2 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{35 c^4}\\ \end{align*}
Mathematica [A] time = 1.63267, size = 385, normalized size = 0.93 \[ \frac{d^3 \left (936 b^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+180 a^2 c^7 x^7+630 a^2 c^6 x^6+756 a^2 c^5 x^5+315 a^2 c^4 x^4+60 a b c^6 x^6+252 a b c^5 x^5+468 a b c^4 x^4+630 a b c^3 x^3+936 a b c^2 x^2+936 a b \log \left (c^2 x^2-1\right )+6 b \tanh ^{-1}(c x) \left (3 a c^4 x^4 \left (20 c^3 x^3+70 c^2 x^2+84 c x+35\right )+b \left (10 c^6 x^6+42 c^5 x^5+78 c^4 x^4+105 c^3 x^3+156 c^2 x^2+315 c x-244\right )-312 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+1890 a b c x+945 a b \log (1-c x)-945 a b \log (c x+1)-1464 a b+12 b^2 c^5 x^5+63 b^2 c^4 x^4+176 b^2 c^3 x^3+441 b^2 c^2 x^2+1386 b^2 \log \left (1-c^2 x^2\right )+9 b^2 \left (20 c^7 x^7+70 c^6 x^6+84 c^5 x^5+35 c^4 x^4-209\right ) \tanh ^{-1}(c x)^2+1464 b^2 c x-504 b^2\right )}{1260 c^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.057, size = 662, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.21887, size = 1253, normalized size = 3.02 \begin{align*} \text{result too large to display} \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 (a^{2} c^{3} d^{3} x^{6} + 3 \, a^{2} c^{2} d^{3} x^{5} + 3 \, a^{2} c d^{3} x^{4} + a^{2} d^{3} x^{3} +{\left (b^{2} c^{3} d^{3} x^{6} + 3 \, b^{2} c^{2} d^{3} x^{5} + 3 \, b^{2} c d^{3} x^{4} + b^{2} d^{3} x^{3}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{3} d^{3} x^{6} + 3 \, a b c^{2} d^{3} x^{5} + 3 \, a b c d^{3} x^{4} + a b d^{3} x^{3}\right )} \operatorname{artanh}\left (c x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d^{3} \left (\int a^{2} x^{3}\, dx + \int 3 a^{2} c x^{4}\, dx + \int 3 a^{2} c^{2} x^{5}\, dx + \int a^{2} c^{3} x^{6}\, dx + \int b^{2} x^{3} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{3} \operatorname{atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{4} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{5} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{6} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{4} \operatorname{atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{5} \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{6} \operatorname{atanh}{\left (c x \right )}\, dx\right ) \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{\left (c d x + d\right )}^{3}{\left (b \operatorname{artanh}\left (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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