Optimal. Leaf size=1795 \[ -\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^4}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right ) b^4}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right ) b^4}{4 a^3 \left (a^2+b^2\right ) d^4}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^4}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b^2}{a^3 d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}-\frac {3 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right ) b^2}{4 a^3 d^4}+\frac {3 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right ) b^2}{4 a^3 d^4}+\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right ) b}{a^2 d}+\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right ) b}{a^2 d^2}+\frac {(e+f x)^3 \text {csch}(c+d x) b}{a^2 d}+\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) b}{a^2 d^3}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right ) b}{a^2 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right ) b}{a^2 d^2}-\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right ) b}{a^2 d^3}-\frac {6 f^3 \text {Li}_3\left (-e^{c+d x}\right ) b}{a^2 d^4}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right ) b}{a^2 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right ) b}{a^2 d^3}+\frac {6 f^3 \text {Li}_3\left (e^{c+d x}\right ) b}{a^2 d^4}-\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right ) b}{a^2 d^4}+\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right ) b}{a^2 d^4}+\frac {(e+f x)^3}{2 a d}-\frac {3 f (e+f x)^2}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {3 f^3 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a d^4}-\frac {3 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.27, antiderivative size = 1795, normalized size of antiderivative = 1.00, number of steps used = 87, number of rules used = 28, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {5589, 2620, 14, 5462, 6741, 12, 6742, 3720, 3716, 2190, 2279, 2391, 32, 2551, 4182, 2531, 6609, 2282, 6589, 2621, 321, 207, 5205, 4180, 5461, 5573, 5561, 3718} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 32
Rule 207
Rule 321
Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 2551
Rule 2620
Rule 2621
Rule 3716
Rule 3718
Rule 3720
Rule 4180
Rule 4182
Rule 5205
Rule 5461
Rule 5462
Rule 5561
Rule 5573
Rule 5589
Rule 6589
Rule 6609
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}-\frac {b \int (e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(3 f) \int (e+f x)^2 \left (-\frac {\coth ^2(c+d x)}{2 d}-\frac {\log (\tanh (c+d x))}{d}\right ) \, dx}{a}\\ &=\frac {b (e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}+\frac {b^2 \int (e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(3 f) \int \frac {(e+f x)^2 \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right )}{2 d} \, dx}{a}+\frac {(3 b f) \int (e+f x)^2 \left (-\frac {\tan ^{-1}(\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a^2}\\ &=\frac {b (e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}+\frac {\left (2 b^2\right ) \int (e+f x)^3 \text {csch}(2 c+2 d x) \, dx}{a^3}-\frac {b^3 \int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(3 b f) \int \frac {(e+f x)^2 \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right )}{d} \, dx}{a^2}-\frac {(3 f) \int (e+f x)^2 \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right ) \, dx}{2 a d}\\ &=\frac {b^4 (e+f x)^4}{4 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}-\frac {b^3 \int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {(3 f) \int \left (-(e+f x)^2 \coth ^2(c+d x)-2 (e+f x)^2 \log (\tanh (c+d x))\right ) \, dx}{2 a d}+\frac {(3 b f) \int (e+f x)^2 \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right ) \, dx}{a^2 d}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^3 d}\\ &=\frac {b^4 (e+f x)^4}{4 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {b^3 \int (e+f x)^3 \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x)^3 \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(3 f) \int (e+f x)^2 \coth ^2(c+d x) \, dx}{2 a d}+\frac {(3 f) \int (e+f x)^2 \log (\tanh (c+d x)) \, dx}{a d}+\frac {(3 b f) \int \left (-(e+f x)^2 \tan ^{-1}(\sinh (c+d x))-(e+f x)^2 \text {csch}(c+d x)\right ) \, dx}{a^2 d}+\frac {\left (3 b^4 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (3 b^4 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (3 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a^3 d^2}-\frac {\left (3 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a^3 d^2}\\ &=-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\int 2 d (e+f x)^3 \text {csch}(2 c+2 d x) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \, dx}{2 a d}-\frac {(3 b f) \int (e+f x)^2 \tan ^{-1}(\sinh (c+d x)) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a^2 d}+\frac {\left (3 i b^3 f\right ) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (3 i b^3 f\right ) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (3 f^2\right ) \int (e+f x) \coth (c+d x) \, dx}{a d^2}+\frac {\left (6 b^4 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac {\left (6 b^4 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}-\frac {\left (3 b^2 f^3\right ) \int \text {Li}_3\left (-e^{2 c+2 d x}\right ) \, dx}{2 a^3 d^3}+\frac {\left (3 b^2 f^3\right ) \int \text {Li}_3\left (e^{2 c+2 d x}\right ) \, dx}{2 a^3 d^3}\\ &=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {2 \int (e+f x)^3 \text {csch}(2 c+2 d x) \, dx}{a}+\frac {b \int d (e+f x)^3 \text {sech}(c+d x) \, dx}{a^2 d}-\frac {\left (3 b^4 f\right ) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}-\frac {\left (6 f^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a d^2}+\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (6 i b^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}+\frac {\left (6 i b^3 f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (3 b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {\left (3 b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a^3 d^4}-\frac {\left (6 b^4 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^3}-\frac {\left (6 b^4 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^3}\\ &=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {6 b f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {3 b^2 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {b \int (e+f x)^3 \text {sech}(c+d x) \, dx}{a^2}+\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {\left (3 b^4 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}-\frac {\left (6 b^4 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {\left (6 b^4 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {\left (3 f^3\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 b f^3\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a^2 d^3}+\frac {\left (6 b f^3\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a^2 d^3}+\frac {\left (6 i b^3 f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (6 i b^3 f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^3}\\ &=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}+\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {6 b f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 b^4 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {3 b^2 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^3 d^4}-\frac {(3 i b f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 d}+\frac {(3 i b f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 d}-\frac {\left (3 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a d^2}+\frac {\left (3 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a d^2}-\frac {\left (3 f^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}-\frac {\left (6 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 i b^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {\left (6 i b^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {\left (3 b^4 f^3\right ) \int \text {Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 a^3 \left (a^2+b^2\right ) d^3}\\ &=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}+\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {6 b f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f^3 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {6 b f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^4}-\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b f^3 \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 b^4 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {6 i b^3 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 i b^3 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {3 b^2 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {\left (6 i b f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (6 i b f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac {\left (3 b^4 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right ) d^4}+\frac {\left (3 f^3\right ) \int \text {Li}_3\left (-e^{2 c+2 d x}\right ) \, dx}{2 a d^3}-\frac {\left (3 f^3\right ) \int \text {Li}_3\left (e^{2 c+2 d x}\right ) \, dx}{2 a d^3}\\ &=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}+\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {6 b f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f^3 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {6 b f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^4}+\frac {6 i b f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 d^3}-\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 d^3}+\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b f^3 \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 b^4 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {6 i b^3 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 i b^3 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}+\frac {3 b^4 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right ) d^4}-\frac {3 b^2 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^3 d^4}+\frac {\left (3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a d^4}-\frac {\left (3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a d^4}-\frac {\left (6 i b f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{a^2 d^3}+\frac {\left (6 i b f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{a^2 d^3}\\ &=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}+\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {6 b f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f^3 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {6 b f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^4}+\frac {6 i b f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 d^3}-\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 d^3}+\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b f^3 \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 b^4 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {6 i b^3 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 i b^3 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}+\frac {3 b^4 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a d^4}-\frac {3 b^2 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^3 d^4}-\frac {3 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^3 d^4}-\frac {\left (6 i b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 i b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}\\ &=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}+\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {6 b f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {3 i b^3 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 b^4 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f^3 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {6 b f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^4}+\frac {6 i b f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 d^3}-\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 d^3}+\frac {6 i b^3 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b f^3 \text {Li}_3\left (e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 b^4 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {6 i b f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a^2 d^4}+\frac {6 i b^3 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 i b f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a^2 d^4}-\frac {6 i b^3 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}+\frac {3 b^4 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a d^4}-\frac {3 b^2 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^3 d^4}-\frac {3 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^3 d^4}\\ \end {align*}
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Mathematica [B] time = 90.00, size = 5823, normalized size = 3.24 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \mathrm {csch}\left (d x +c \right )^{3} \mathrm {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]
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 \[ \int \frac {{\left (e+f\,x\right )}^3}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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