Optimal. Leaf size=448 \[ \frac {3 b c \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {3 b c \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}+\frac {3 b c \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {9 b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (c x+1)}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (c x+1)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (c x+1)}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (c x+1)^2}+\frac {17 c \left (a+b \tanh ^{-1}(c x)\right )^2}{8 d^3}-\frac {6 c \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {3 c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3}-\frac {b^2 c \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{d^3}-\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (\frac {2}{1-c x}-1\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 d^3}-\frac {19 b^2 c}{16 d^3 (c x+1)}-\frac {b^2 c}{16 d^3 (c x+1)^2}+\frac {19 b^2 c \tanh ^{-1}(c x)}{16 d^3} \]
[Out]
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Rubi [A] time = 0.98, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 17, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {5940, 5916, 5988, 5932, 2447, 5914, 6052, 5948, 6058, 6610, 5928, 5926, 627, 44, 207, 5918, 6056} \[ \frac {3 b c \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {3 b c \text {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}+\frac {3 b c \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {b^2 c \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d^3}-\frac {3 b^2 c \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^3}+\frac {3 b^2 c \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^3}-\frac {9 b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (c x+1)}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (c x+1)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (c x+1)}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (c x+1)^2}+\frac {17 c \left (a+b \tanh ^{-1}(c x)\right )^2}{8 d^3}-\frac {6 c \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^3}-\frac {3 c \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3}-\frac {19 b^2 c}{16 d^3 (c x+1)}-\frac {b^2 c}{16 d^3 (c x+1)^2}+\frac {19 b^2 c \tanh ^{-1}(c x)}{16 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 207
Rule 627
Rule 2447
Rule 5914
Rule 5916
Rule 5918
Rule 5926
Rule 5928
Rule 5932
Rule 5940
Rule 5948
Rule 5988
Rule 6052
Rule 6056
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2 (d+c d x)^3} \, dx &=\int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x^2}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)^3}+\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx}{d^3}-\frac {(3 c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx}{d^3}+\frac {c^2 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^3} \, dx}{d^3}+\frac {\left (2 c^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{d^3}+\frac {\left (3 c^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{d^3}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}-\frac {6 c \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {(2 b c) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d^3}+\frac {\left (b c^2\right ) \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}{d^3}+\frac {\left (4 b c^2\right ) \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}{d^3}+\frac {\left (6 b c^2\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}{d^3}+\frac {\left (12 b c^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^3}\\ &=\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}-\frac {6 c \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^3}+\frac {(2 b c) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx}{d^3}+\frac {\left (b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{4 d^3}-\frac {\left (b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{4 d^3}+\frac {\left (b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{2 d^3}+\frac {\left (2 b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{d^3}-\frac {\left (2 b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{d^3}-\frac {\left (6 b c^2\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}{d^3}+\frac {\left (6 b c^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^3}-\frac {\left (3 b^2 c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^3}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)^2}-\frac {9 b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)}+\frac {17 c \left (a+b \tanh ^{-1}(c x)\right )^2}{8 d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}-\frac {6 c \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {2 b c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{4 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{4 d^3}+\frac {\left (2 b^2 c^2\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{d^3}-\frac {\left (2 b^2 c^2\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^3}-\frac {\left (3 b^2 c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^3}+\frac {\left (3 b^2 c^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^3}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)^2}-\frac {9 b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)}+\frac {17 c \left (a+b \tanh ^{-1}(c x)\right )^2}{8 d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}-\frac {6 c \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {2 b c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^3}-\frac {b^2 c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^3}-\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{(1-c x) (1+c x)^3} \, dx}{4 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{4 d^3}+\frac {\left (2 b^2 c^2\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{d^3}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)^2}-\frac {9 b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)}+\frac {17 c \left (a+b \tanh ^{-1}(c x)\right )^2}{8 d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}-\frac {6 c \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {2 b c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^3}-\frac {b^2 c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^3}-\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 d^3}+\frac {\left (b^2 c^2\right ) \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 d^3}+\frac {\left (2 b^2 c^2\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d^3}\\ &=-\frac {b^2 c}{16 d^3 (1+c x)^2}-\frac {19 b^2 c}{16 d^3 (1+c x)}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)^2}-\frac {9 b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)}+\frac {17 c \left (a+b \tanh ^{-1}(c x)\right )^2}{8 d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}-\frac {6 c \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {2 b c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^3}-\frac {b^2 c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^3}-\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^3}-\frac {\left (b^2 c^2\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{16 d^3}-\frac {\left (b^2 c^2\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{8 d^3}-\frac {\left (b^2 c^2\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{d^3}\\ &=-\frac {b^2 c}{16 d^3 (1+c x)^2}-\frac {19 b^2 c}{16 d^3 (1+c x)}+\frac {19 b^2 c \tanh ^{-1}(c x)}{16 d^3}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)^2}-\frac {9 b c \left (a+b \tanh ^{-1}(c x)\right )}{4 d^3 (1+c x)}+\frac {17 c \left (a+b \tanh ^{-1}(c x)\right )^2}{8 d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^3 (1+c x)^2}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^3 (1+c x)}-\frac {6 c \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 c \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^3}+\frac {2 b c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^3}-\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^3}+\frac {3 b c \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^3}-\frac {b^2 c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^3}-\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^3}+\frac {3 b^2 c \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^3}\\ \end {align*}
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Mathematica [C] time = 2.35, size = 479, normalized size = 1.07 \[ \frac {-\frac {128 a^2 c}{c x+1}-\frac {32 a^2 c}{(c x+1)^2}-192 a^2 c \log (x)+192 a^2 c \log (c x+1)-\frac {64 a^2}{x}+\frac {4 a b \left (c x \left (32 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+20 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )-20 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )\right )+48 c x \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )-4 \tanh ^{-1}(c x) \left (24 c x \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-10 c x \sinh \left (2 \tanh ^{-1}(c x)\right )-c x \sinh \left (4 \tanh ^{-1}(c x)\right )+10 c x \cosh \left (2 \tanh ^{-1}(c x)\right )+c x \cosh \left (4 \tanh ^{-1}(c x)\right )+8\right )\right )}{x}+b^2 c \left (-192 \tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )-64 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+96 \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )+128 \tanh ^{-1}(c x)^3-\frac {64 \tanh ^{-1}(c x)^2}{c x}+64 \tanh ^{-1}(c x)^2-192 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+128 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+80 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )+8 \tanh ^{-1}(c x)^2 \sinh \left (4 \tanh ^{-1}(c x)\right )+80 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \sinh \left (4 \tanh ^{-1}(c x)\right )+40 \sinh \left (2 \tanh ^{-1}(c x)\right )+\sinh \left (4 \tanh ^{-1}(c x)\right )-80 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )-8 \tanh ^{-1}(c x)^2 \cosh \left (4 \tanh ^{-1}(c x)\right )-80 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )-4 \tanh ^{-1}(c x) \cosh \left (4 \tanh ^{-1}(c x)\right )-40 \cosh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (4 \tanh ^{-1}(c x)\right )-8 i \pi ^3\right )}{64 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}}{c^{3} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{4} + 3 \, c d^{3} x^{3} + d^{3} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.04, size = 7593, normalized size = 16.95 \[ \text {output too large to display} \]
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 \[ -\frac {1}{2} \, a^{2} {\left (\frac {6 \, c^{2} x^{2} + 9 \, c x + 2}{c^{2} d^{3} x^{3} + 2 \, c d^{3} x^{2} + d^{3} x} - \frac {6 \, c \log \left (c x + 1\right )}{d^{3}} + \frac {6 \, c \log \relax (x)}{d^{3}}\right )} - \frac {{\left (6 \, b^{2} c^{2} x^{2} + 9 \, b^{2} c x + 2 \, b^{2} - 6 \, {\left (b^{2} c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} + b^{2} c x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{8 \, {\left (c^{2} d^{3} x^{3} + 2 \, c d^{3} x^{2} + d^{3} x\right )}} - \int -\frac {{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \, {\left (a b c x - a b\right )} \log \left (c x + 1\right ) + {\left (6 \, b^{2} c^{4} x^{4} + 15 \, b^{2} c^{3} x^{3} + 11 \, b^{2} c^{2} x^{2} + 4 \, a b - 2 \, {\left (2 \, a b c - b^{2} c\right )} x - 2 \, {\left (3 \, b^{2} c^{5} x^{5} + 9 \, b^{2} c^{4} x^{4} + 9 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} + b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \, {\left (c^{4} d^{3} x^{6} + 2 \, c^{3} d^{3} x^{5} - 2 \, c d^{3} x^{3} - d^{3} x^{2}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d+c\,d\,x\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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