Optimal. Leaf size=480 \[ -\frac {3 b c^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {3 b c^2 \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {3 b c^2 \text {Li}_2\left (1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (c x+1)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}+\frac {6 c^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {3 c^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {4 b c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {2 b^2 c^2 \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{d^2}+\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (\frac {2}{1-c x}-1\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{c x+1}\right )}{2 d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {b^2 c^2}{2 d^2 (c x+1)}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \tanh ^{-1}(c x)}{2 d^2} \]
[Out]
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Rubi [A] time = 0.95, antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 22, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5940, 5916, 5982, 266, 36, 29, 31, 5948, 5988, 5932, 2447, 5914, 6052, 6058, 6610, 5928, 5926, 627, 44, 207, 5918, 6056} \[ -\frac {3 b c^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {3 b c^2 \text {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {3 b c^2 \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac {2 b^2 c^2 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d^2}+\frac {3 b^2 c^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^2}-\frac {3 b^2 c^2 \text {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^2}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (c x+1)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}+\frac {6 c^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac {3 c^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {4 b c^2 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {b^2 c^2}{2 d^2 (c x+1)}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \tanh ^{-1}(c x)}{2 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 44
Rule 207
Rule 266
Rule 627
Rule 2447
Rule 5914
Rule 5916
Rule 5918
Rule 5926
Rule 5928
Rule 5932
Rule 5940
Rule 5948
Rule 5982
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^3 (d+c d x)^2} \, dx &=\int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x^3}-\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}-\frac {c^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)^2}-\frac {3 c^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx}{d^2}-\frac {(2 c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx}{d^2}+\frac {\left (3 c^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx}{d^2}-\frac {c^3 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx}{d^2}-\frac {\left (3 c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1+c x} \, dx}{d^2}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {(b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx}{d^2}-\frac {\left (4 b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d^2}-\frac {\left (2 b c^3\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^2}-\frac {\left (6 b 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}{d^2}-\frac {\left (12 b c^3\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^2}\\ &=-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {(b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d^2}-\frac {\left (4 b c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx}{d^2}-\frac {\left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{d^2}+\frac {\left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{d^2}+\frac {\left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{d^2}+\frac {\left (6 b 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}{d^2}-\frac {\left (6 b c^3\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^2}+\frac {\left (3 b^2 c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {4 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{d^2}-\frac {\left (b^2 c^3\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{d^2}+\frac {\left (3 b^2 c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}-\frac {\left (3 b^2 c^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{d^2}+\frac {\left (4 b^2 c^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d^2}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {4 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b^2 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^2}+\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (b^2 c^3\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{d^2}\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {4 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b^2 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^2}+\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (b^2 c^3\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d^2}+\frac {\left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )}{2 d^2}\\ &=\frac {b^2 c^2}{2 d^2 (1+c x)}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {4 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b^2 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^2}+\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}+\frac {\left (b^2 c^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac {b^2 c^2}{2 d^2 (1+c x)}-\frac {b^2 c^2 \tanh ^{-1}(c x)}{2 d^2}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {4 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b^2 c^2 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{d^2}+\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {Li}_3\left (1-\frac {2}{1+c x}\right )}{2 d^2}\\ \end {align*}
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Mathematica [C] time = 2.17, size = 452, normalized size = 0.94 \[ \frac {\frac {8 a^2 c^2}{c x+1}+24 a^2 c^2 \log (x)-24 a^2 c^2 \log (c x+1)+\frac {16 a^2 c}{x}-\frac {4 a^2}{x^2}+\frac {4 a b \left (-6 c^2 x^2 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )+c x \left (-8 c x \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-c x \sinh \left (2 \tanh ^{-1}(c x)\right )+c x \cosh \left (2 \tanh ^{-1}(c x)\right )-2\right )+2 \tanh ^{-1}(c x) \left (c^2 x^2+6 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-c^2 x^2 \sinh \left (2 \tanh ^{-1}(c x)\right )+c^2 x^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+4 c x-1\right )\right )}{x^2}+b^2 c^2 \left (8 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-\frac {4 \tanh ^{-1}(c x)^2}{c^2 x^2}+24 \tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )+16 \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )-12 \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )-16 \tanh ^{-1}(c x)^3+\frac {16 \tanh ^{-1}(c x)^2}{c x}-12 \tanh ^{-1}(c x)^2-\frac {8 \tanh ^{-1}(c x)}{c x}+24 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )-32 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-4 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )-4 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )-2 \sinh \left (2 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )+2 \cosh \left (2 \tanh ^{-1}(c x)\right )+i \pi ^3\right )}{8 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x\right ) + a^{2}}{c^{2} d^{2} x^{5} + 2 \, c d^{2} x^{4} + d^{2} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.34, size = 2009, normalized size = 4.19 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, a^{2} {\left (\frac {6 \, c^{2} \log \left (c x + 1\right )}{d^{2}} - \frac {6 \, c^{2} \log \relax (x)}{d^{2}} - \frac {6 \, c^{2} x^{2} + 3 \, c x - 1}{c d^{2} x^{3} + d^{2} x^{2}}\right )} + \frac {{\left (6 \, b^{2} c^{2} x^{2} + 3 \, b^{2} c x - b^{2} - 6 \, {\left (b^{2} c^{3} x^{3} + b^{2} c^{2} x^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{8 \, {\left (c d^{2} x^{3} + d^{2} x^{2}\right )}} + \int \frac {{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \, {\left (a b c x - a b\right )} \log \left (c x + 1\right ) - {\left (6 \, b^{2} c^{4} x^{4} + 9 \, b^{2} c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 4 \, a b + {\left (4 \, a b c - b^{2} c\right )} x - 2 \, {\left (3 \, b^{2} c^{5} x^{5} + 6 \, b^{2} c^{4} x^{4} + 3 \, b^{2} c^{3} x^{3} - b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \, {\left (c^{3} d^{2} x^{6} + c^{2} d^{2} x^{5} - c d^{2} x^{4} - d^{2} x^{3}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d+c\,d\,x\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{5} + 2 c x^{4} + x^{3}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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