Optimal. Leaf size=1049 \[ \frac {6 i b \text {Li}_4\left (-i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac {6 i b \text {Li}_4\left (i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}-\frac {6 b^2 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}+\frac {3 b^2 \text {Li}_4\left (-e^{2 (c+d x)}\right ) f^3}{4 a \left (a^2+b^2\right ) d^4}-\frac {3 \text {Li}_4\left (-e^{2 c+2 d x}\right ) f^3}{4 a d^4}+\frac {3 \text {Li}_4\left (e^{2 c+2 d x}\right ) f^3}{4 a d^4}-\frac {6 i b (e+f x) \text {Li}_3\left (-i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac {6 i b (e+f x) \text {Li}_3\left (i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right ) f^2}{2 a \left (a^2+b^2\right ) d^3}+\frac {3 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right ) f^2}{2 a d^3}-\frac {3 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right ) f^2}{2 a d^3}+\frac {3 i b (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac {3 i b (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) f}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right ) f}{2 a d^2}+\frac {3 (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right ) f}{2 a d^2}-\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.34, antiderivative size = 1049, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {5589, 5461, 4182, 2531, 6609, 2282, 6589, 5573, 5561, 2190, 6742, 4180, 3718} \[ \frac {6 i b \text {PolyLog}\left (4,-i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac {6 i b \text {PolyLog}\left (4,i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}-\frac {6 b^2 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}+\frac {3 b^2 \text {PolyLog}\left (4,-e^{2 (c+d x)}\right ) f^3}{4 a \left (a^2+b^2\right ) d^4}-\frac {3 \text {PolyLog}\left (4,-e^{2 c+2 d x}\right ) f^3}{4 a d^4}+\frac {3 \text {PolyLog}\left (4,e^{2 c+2 d x}\right ) f^3}{4 a d^4}-\frac {6 i b (e+f x) \text {PolyLog}\left (3,-i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac {6 i b (e+f x) \text {PolyLog}\left (3,i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^2 (e+f x) \text {PolyLog}\left (3,-e^{2 (c+d x)}\right ) f^2}{2 a \left (a^2+b^2\right ) d^3}+\frac {3 (e+f x) \text {PolyLog}\left (3,-e^{2 c+2 d x}\right ) f^2}{2 a d^3}-\frac {3 (e+f x) \text {PolyLog}\left (3,e^{2 c+2 d x}\right ) f^2}{2 a d^3}+\frac {3 i b (e+f x)^2 \text {PolyLog}\left (2,-i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac {3 i b (e+f x)^2 \text {PolyLog}\left (2,i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 (e+f x)^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) f}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 (e+f x)^2 \text {PolyLog}\left (2,-e^{2 c+2 d x}\right ) f}{2 a d^2}+\frac {3 (e+f x)^2 \text {PolyLog}\left (2,e^{2 c+2 d x}\right ) f}{2 a d^2}-\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 4180
Rule 4182
Rule 5461
Rule 5561
Rule 5573
Rule 5589
Rule 6589
Rule 6609
Rule 6742
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {2 \int (e+f x)^3 \text {csch}(2 c+2 d x) \, dx}{a}-\frac {b \int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {b^2 (e+f x)^4}{4 a \left (a^2+b^2\right ) f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b \int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\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 {b^2 (e+f x)^4}{4 a \left (a^2+b^2\right ) f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\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 {b \int (e+f x)^3 \text {sech}(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^3 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}+\frac {\left (3 b^2 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 \left (a^2+b^2\right ) d}+\frac {\left (3 b^2 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 \left (a^2+b^2\right ) 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 {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \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 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 {\left (2 b^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )}+\frac {(3 i b f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac {(3 i b f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (6 b^2 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 \left (a^2+b^2\right ) d^2}+\frac {\left (6 b^2 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 \left (a^2+b^2\right ) d^2}-\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 {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \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 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \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 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d}-\frac {\left (6 i b f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) 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}{\left (a^2+b^2\right ) d^2}-\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 b^2 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}-\frac {\left (6 b^2 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}\\ &=-\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \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 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \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 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}-\frac {\left (3 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}-\frac {\left (6 b^2 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 \left (a^2+b^2\right ) d^4}-\frac {\left (6 b^2 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 \left (a^2+b^2\right ) d^4}+\frac {\left (6 i b f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3}-\frac {\left (6 i b f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3}\\ &=-\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \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 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \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 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \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 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a 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 )}{\left (a^2+b^2\right ) 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 )}{\left (a^2+b^2\right ) d^4}+\frac {\left (3 b^2 f^3\right ) \int \text {Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 a \left (a^2+b^2\right ) d^3}\\ &=-\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \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 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \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 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}+\frac {6 i b f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 i b f^3 \text {Li}_4\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \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 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a 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+d x)}\right )}{4 a \left (a^2+b^2\right ) d^4}\\ &=-\frac {2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \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 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{\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 )}{\left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^2 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \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 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}+\frac {6 i b f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 i b f^3 \text {Li}_4\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 a \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 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4}\\ \end {align*}
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Mathematica [B] time = 34.48, size = 3872, normalized size = 3.69 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.67, size = 2454, normalized size = 2.34 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]
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 \[ -e^{3} {\left (\frac {b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{3} + a b^{2}\right )} d} - \frac {2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} - \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} + \frac {3 \, {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )} e^{2} f}{a d^{2}} + \frac {3 \, {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )} e^{2} f}{a d^{2}} + \frac {3 \, {\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} e f^{2}}{a d^{3}} + \frac {3 \, {\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} e f^{2}}{a d^{3}} + \frac {{\left (d^{3} x^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(-e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} + \frac {{\left (d^{3} x^{3} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} - \frac {d^{4} f^{3} x^{4} + 4 \, d^{4} e f^{2} x^{3} + 6 \, d^{4} e^{2} f x^{2}}{2 \, a d^{4}} + \int \frac {2 \, {\left (b^{3} f^{3} x^{3} + 3 \, b^{3} e f^{2} x^{2} + 3 \, b^{3} e^{2} f x - {\left (a b^{2} f^{3} x^{3} e^{c} + 3 \, a b^{2} e f^{2} x^{2} e^{c} + 3 \, a b^{2} e^{2} f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{3} b + a b^{3} - {\left (a^{3} b e^{\left (2 \, c\right )} + a b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{4} e^{c} + a^{2} b^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int -\frac {2 \, {\left (a f^{3} x^{3} + 3 \, a e f^{2} x^{2} + 3 \, a e^{2} f x - {\left (b f^{3} x^{3} e^{c} + 3 \, b e f^{2} x^{2} e^{c} + 3 \, b e^{2} f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} + b^{2} + {\left (a^{2} e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\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 \[ \int \frac {{\left (e+f\,x\right )}^3}{\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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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