Optimal. Leaf size=953 \[ -\frac {3 c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) b^3}{4 (c d-e) (c d+e)^2}-\frac {3 c^2 \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^3}{4 (c d-e)^2 (c d+e)}+\frac {3 c^2 e \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right ) b^3}{8 e (c d+e)^2}+\frac {3 c^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right ) b^3}{8 (c d-e)^2 e}-\frac {3 c^3 d \text {Li}_3\left (1-\frac {2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}+\frac {3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right ) b^2}{2 (c d-e) (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{c x+1}\right ) b^2}{2 (c d-e)^2 (c d+e)}-\frac {3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right ) b^2}{4 e (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^2}{4 (c d-e)^2 e}-\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 b}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right ) b}{4 e (c d+e)^2}-\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{c x+1}\right ) b}{4 (c d-e)^2 e}+\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{c x+1}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.03, antiderivative size = 953, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5928, 5918, 5948, 6058, 6610, 6056, 2402, 2315, 5920, 2447, 5922} \[ -\frac {3 c^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) b^3}{4 (c d-e) (c d+e)^2}-\frac {3 c^2 \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b^3}{4 (c d-e)^2 (c d+e)}+\frac {3 c^2 e \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 e \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right ) b^3}{8 e (c d+e)^2}+\frac {3 c^2 \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right ) b^3}{8 (c d-e)^2 e}-\frac {3 c^3 d \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}+\frac {3 c^3 d \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right ) b^2}{2 (c d-e) (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{c x+1}\right ) b^2}{2 (c d-e)^2 (c d+e)}-\frac {3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) b^2}{4 e (c d+e)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b^2}{4 (c d-e)^2 e}-\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 b}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right ) b}{4 e (c d+e)^2}-\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{c x+1}\right ) b}{4 (c d-e)^2 e}+\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{c x+1}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac {3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2315
Rule 2402
Rule 2447
Rule 5918
Rule 5920
Rule 5922
Rule 5928
Rule 5948
Rule 6056
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{(d+e x)^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {(3 b c) \int \left (-\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d+e)^2 (-1+c x)}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e)^2 (1+c x)}+\frac {e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{(-c d+e) (c d+e) (d+e x)^2}-\frac {2 c^2 d e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{(c d-e)^2 (c d+e)^2 (d+e x)}\right ) \, dx}{2 e}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {\left (3 b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{4 (c d-e)^2 e}-\frac {\left (3 b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c x} \, dx}{4 e (c d+e)^2}-\frac {\left (3 b c^3 d e\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d+e x} \, dx}{(c d-e)^2 (c d+e)^2}+\frac {(3 b c e) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx}{2 (-c d+e) (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {\left (3 b^2 c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 (c d-e)^2 e}-\frac {\left (3 b^2 c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{2 e (c d+e)^2}+\frac {\left (3 b^2 c^2\right ) \int \left (-\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 (c d+e) (-1+c x)}+\frac {c \left (a+b \tanh ^{-1}(c x)\right )}{2 (c d-e) (1+c x)}+\frac {e^2 \left (a+b \tanh ^{-1}(c x)\right )}{(-c d+e) (c d+e) (d+e x)}\right ) \, dx}{(-c d+e) (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{4 (c d-e)^2 e}+\frac {\left (3 b^2 c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c x} \, dx}{2 (c d-e) (c d+e)^2}-\frac {\left (3 b^3 c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{4 e (c d+e)^2}+\frac {\left (3 b^2 c^2 e^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^2 c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{2 (c d-e)^2 (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {\left (3 b^3 c^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{2 (c d-e) (c d+e)^2}+\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^3 e\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 (c d-e)^2 (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {\left (3 b^3 c^2 e\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {\left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}\\ &=\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 (c d-e) (c d+e) (d+e x)}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 (c d-e) (c d+e)^2}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^3 c^2 e \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 (c d+e)}+\frac {3 b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}\\ \end {align*}
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Mathematica [F] time = 85.89, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{(d+e x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c x\right ) + a^{3}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.86, size = 53538, normalized size = 56.18 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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