Optimal. Leaf size=480 \[ \frac {b^2 c^2}{2 d^2 (1+c x)}-\frac {b^2 c^2 \tanh ^{-1}(c x)}{2 d^2}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 x}+\frac {b c^2 \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (1+c x)}-\frac {2 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2 x^2}+\frac {2 c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}+\frac {c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (1+c x)}+\frac {6 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}+\frac {3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {4 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^2}+\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^2}-\frac {3 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^2}+\frac {2 b^2 c^2 \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d^2}+\frac {3 b^2 c^2 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^2}-\frac {3 b^2 c^2 \text {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^2} \]
[Out]
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Rubi [A]
time = 0.69, 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 = {6087, 6037,
6129, 272, 36, 29, 31, 6095, 6135, 6079, 2497, 6033, 6199, 6205, 6745, 6065, 6063, 641, 46, 213,
6055, 6203} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 46
Rule 213
Rule 272
Rule 641
Rule 2497
Rule 6033
Rule 6037
Rule 6055
Rule 6063
Rule 6065
Rule 6079
Rule 6087
Rule 6095
Rule 6129
Rule 6135
Rule 6199
Rule 6203
Rule 6205
Rule 6745
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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 1.35, size = 452, normalized size = 0.94 \begin {gather*} \frac {-\frac {4 a^2}{x^2}+\frac {16 a^2 c}{x}+\frac {8 a^2 c^2}{1+c x}+24 a^2 c^2 \log (x)-24 a^2 c^2 \log (1+c x)+b^2 c^2 \left (i \pi ^3-\frac {8 \tanh ^{-1}(c x)}{c x}-12 \tanh ^{-1}(c x)^2-\frac {4 \tanh ^{-1}(c x)^2}{c^2 x^2}+\frac {16 \tanh ^{-1}(c x)^2}{c x}-16 \tanh ^{-1}(c x)^3+2 \cosh \left (2 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )+4 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )-32 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+24 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+8 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+16 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+24 \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )-12 \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )-2 \sinh \left (2 \tanh ^{-1}(c x)\right )-4 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )-4 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )\right )+\frac {4 a b \left (-6 c^2 x^2 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+c x \left (-2+c x \cosh \left (2 \tanh ^{-1}(c x)\right )-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 )\right )+2 \tanh ^{-1}(c x) \left (-1+4 c x+c^2 x^2+c^2 x^2 \cosh \left (2 \tanh ^{-1}(c x)\right )+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 )\right )\right )}{x^2}}{8 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 14.03, size = 1878, normalized size = 3.91
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1878\) |
default | \(\text {Expression too large to display}\) | \(1878\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d+c\,d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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