Optimal. Leaf size=408 \[ \frac{6 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3}+\frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^5 d^3}+\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{c^5 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}+\frac{a b x}{c^4 d^3}+\frac{15 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (c x+1)^2}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (c x+1)^2}-\frac{43 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^5 d^3}+\frac{6 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3}-\frac{6 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac{29 b^2}{16 c^5 d^3 (c x+1)}-\frac{b^2}{16 c^5 d^3 (c x+1)^2}+\frac{b^2 x \tanh ^{-1}(c x)}{c^4 d^3}-\frac{29 b^2 \tanh ^{-1}(c x)}{16 c^5 d^3} \]
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Rubi [A] time = 0.810864, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {5940, 5910, 5984, 5918, 2402, 2315, 5916, 5980, 260, 5948, 5928, 5926, 627, 44, 207, 6056, 6610} \[ \frac{6 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3}+\frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{c^5 d^3}+\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{c^5 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}+\frac{a b x}{c^4 d^3}+\frac{15 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (c x+1)^2}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (c x+1)^2}-\frac{43 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^5 d^3}+\frac{6 b \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^5 d^3}-\frac{6 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac{29 b^2}{16 c^5 d^3 (c x+1)}-\frac{b^2}{16 c^5 d^3 (c x+1)^2}+\frac{b^2 x \tanh ^{-1}(c x)}{c^4 d^3}-\frac{29 b^2 \tanh ^{-1}(c x)}{16 c^5 d^3} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 5916
Rule 5980
Rule 260
Rule 5948
Rule 5928
Rule 5926
Rule 627
Rule 44
Rule 207
Rule 6056
Rule 6610
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^3} \, dx &=\int \left (-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3 d^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3 (1+c x)^3}-\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3 (1+c x)^2}+\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3 (1+c x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx}{c^4 d^3}-\frac{3 \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^4 d^3}-\frac{4 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{c^4 d^3}+\frac{6 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{c^4 d^3}+\frac{\int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c^3 d^3}\\ &=-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (1+c x)^2}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (1+c x)}-\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{b \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^3}+\frac{a+b \tanh ^{-1}(c x)}{4 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}-\frac{(8 b) \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}+\frac{(12 b) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^3}+\frac{(6 b) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^3 d^3}-\frac{b \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^2 d^3}\\ &=-\frac{3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (1+c x)^2}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (1+c x)}-\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{4 c^4 d^3}-\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{4 c^4 d^3}+\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{2 c^4 d^3}+\frac{b \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^4 d^3}-\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^4 d^3}-\frac{(4 b) \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^4 d^3}+\frac{(4 b) \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^4 d^3}+\frac{(6 b) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^4 d^3}-\frac{\left (6 b^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^3}\\ &=\frac{a b x}{c^4 d^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)^2}+\frac{15 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)}-\frac{43 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (1+c x)^2}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (1+c x)}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^5 d^3}-\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{3 b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{b^2 \int \frac{1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{4 c^4 d^3}+\frac{b^2 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{4 c^4 d^3}+\frac{b^2 \int \tanh ^{-1}(c x) \, dx}{c^4 d^3}-\frac{\left (4 b^2\right ) \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^4 d^3}-\frac{\left (6 b^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^4 d^3}\\ &=\frac{a b x}{c^4 d^3}+\frac{b^2 x \tanh ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)^2}+\frac{15 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)}-\frac{43 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (1+c x)^2}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (1+c x)}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^5 d^3}-\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{3 b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{c^5 d^3}+\frac{b^2 \int \frac{1}{(1-c x) (1+c x)^3} \, dx}{4 c^4 d^3}+\frac{b^2 \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{4 c^4 d^3}-\frac{\left (4 b^2\right ) \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{c^4 d^3}-\frac{b^2 \int \frac{x}{1-c^2 x^2} \, dx}{c^3 d^3}\\ &=\frac{a b x}{c^4 d^3}+\frac{b^2 x \tanh ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)^2}+\frac{15 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)}-\frac{43 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (1+c x)^2}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (1+c x)}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^5 d^3}-\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac{3 b^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^5 d^3}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{3 b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{b^2 \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 c^4 d^3}+\frac{b^2 \int \left (\frac{1}{2 (1+c x)^3}+\frac{1}{4 (1+c x)^2}-\frac{1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 c^4 d^3}-\frac{\left (4 b^2\right ) \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c^4 d^3}\\ &=\frac{a b x}{c^4 d^3}-\frac{b^2}{16 c^5 d^3 (1+c x)^2}+\frac{29 b^2}{16 c^5 d^3 (1+c x)}+\frac{b^2 x \tanh ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)^2}+\frac{15 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)}-\frac{43 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (1+c x)^2}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (1+c x)}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^5 d^3}-\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac{3 b^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^5 d^3}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{3 b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}-\frac{b^2 \int \frac{1}{-1+c^2 x^2} \, dx}{16 c^4 d^3}-\frac{b^2 \int \frac{1}{-1+c^2 x^2} \, dx}{8 c^4 d^3}+\frac{\left (2 b^2\right ) \int \frac{1}{-1+c^2 x^2} \, dx}{c^4 d^3}\\ &=\frac{a b x}{c^4 d^3}-\frac{b^2}{16 c^5 d^3 (1+c x)^2}+\frac{29 b^2}{16 c^5 d^3 (1+c x)}-\frac{29 b^2 \tanh ^{-1}(c x)}{16 c^5 d^3}+\frac{b^2 x \tanh ^{-1}(c x)}{c^4 d^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)^2}+\frac{15 b \left (a+b \tanh ^{-1}(c x)\right )}{4 c^5 d^3 (1+c x)}-\frac{43 \left (a+b \tanh ^{-1}(c x)\right )^2}{8 c^5 d^3}-\frac{3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^4 d^3}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3 d^3}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5 d^3 (1+c x)^2}+\frac{4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^5 d^3 (1+c x)}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{c^5 d^3}-\frac{6 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{b^2 \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac{3 b^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{c^5 d^3}+\frac{6 b \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}+\frac{3 b^2 \text{Li}_3\left (1-\frac{2}{1+c x}\right )}{c^5 d^3}\\ \end{align*}
Mathematica [A] time = 2.01257, size = 420, normalized size = 1.03 \[ \frac{a b \left (96 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-48 \log \left (1-c^2 x^2\right )+4 \tanh ^{-1}(c x) \left (4 c^2 x^2-24 c x-48 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-14 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+14 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )-4\right )+16 c x-28 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+28 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )\right )+16 b^2 \left (\frac{1}{64} \left (192 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )+32 \log \left (1-c^2 x^2\right )+8 \tanh ^{-1}(c x)^2 \left (4 c^2 x^2-24 c x-48 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-14 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+14 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )+20\right )-56 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+56 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \left (16 c x+96 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-28 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )+28 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )\right )\right )+\left (6 \tanh ^{-1}(c x)-3\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )+8 a^2 c^2 x^2-48 a^2 c x+\frac{64 a^2}{c x+1}-\frac{8 a^2}{(c x+1)^2}+96 a^2 \log (c x+1)}{16 c^5 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.816, size = 1565, normalized size = 3.8 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{8 \, c x + 7}{c^{7} d^{3} x^{2} + 2 \, c^{6} d^{3} x + c^{5} d^{3}} + \frac{c x^{2} - 6 \, x}{c^{4} d^{3}} + \frac{12 \, \log \left (c x + 1\right )}{c^{5} d^{3}}\right )} + \frac{{\left (b^{2} c^{4} x^{4} - 4 \, b^{2} c^{3} x^{3} - 11 \, b^{2} c^{2} x^{2} + 2 \, b^{2} c x + 7 \, b^{2} + 12 \,{\left (b^{2} c^{2} x^{2} + 2 \, b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{8 \,{\left (c^{7} d^{3} x^{2} + 2 \, c^{6} d^{3} x + c^{5} d^{3}\right )}} - \int -\frac{{\left (b^{2} c^{5} x^{5} - b^{2} c^{4} x^{4}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c^{5} x^{5} - a b c^{4} x^{4}\right )} \log \left (c x + 1\right ) +{\left (15 \, b^{2} c^{3} x^{3} + 9 \, b^{2} c^{2} x^{2} -{\left (4 \, a b c^{5} + b^{2} c^{5}\right )} x^{5} +{\left (4 \, a b c^{4} + 3 \, b^{2} c^{4}\right )} x^{4} - 9 \, b^{2} c x - 7 \, b^{2} - 2 \,{\left (b^{2} c^{5} x^{5} - b^{2} c^{4} x^{4} + 6 \, b^{2} c^{3} x^{3} + 18 \, b^{2} c^{2} x^{2} + 18 \, b^{2} c x + 6 \, b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c^{8} d^{3} x^{4} + 2 \, c^{7} d^{3} x^{3} - 2 \, c^{5} d^{3} x - c^{4} d^{3}\right )}}\,{d x} \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} x^{4} \operatorname{artanh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname{artanh}\left (c x\right ) + a^{2} x^{4}}{c^{3} d^{3} x^{3} + 3 \, c^{2} d^{3} x^{2} + 3 \, c d^{3} x + d^{3}}, 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*} \frac{\int \frac{a^{2} x^{4}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac{b^{2} x^{4} \operatorname{atanh}^{2}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac{2 a b x^{4} \operatorname{atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \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\right ) + a\right )}^{2} x^{4}}{{\left (c d x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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