Optimal. Leaf size=1118 \[ \frac {(e+f x)^3 a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {3 f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right ) a^3}{2 b^2 \left (a^2+b^2\right ) d^4}+\frac {(e+f x)^3 \tanh (c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}+\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^{3/2} d}-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^{3/2} d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} 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 ) a^2}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} 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 ) a^2}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x) a^2}{b \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 a}{b^2 d}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{b^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) a}{b^2 d^3}-\frac {3 f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right ) a}{2 b^2 d^4}-\frac {(e+f x)^3 \tanh (c+d x) a}{b^2 d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.98, antiderivative size = 1118, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 15, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5567, 5451, 4180, 2531, 2282, 6589, 5583, 4184, 3718, 2190, 5573, 3322, 2264, 6609, 6742} \[ \frac {(e+f x)^3 a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {3 f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right ) a^3}{2 b^2 \left (a^2+b^2\right ) d^4}+\frac {(e+f x)^3 \tanh (c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}+\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^{3/2} d}-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^{3/2} d}+\frac {6 i f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {6 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {PolyLog}\left (3,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x) a^2}{b \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 a}{b^2 d}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{b^2 d^2}+\frac {3 f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a}{b^2 d^3}-\frac {3 f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right ) a}{2 b^2 d^4}-\frac {(e+f x)^3 \tanh (c+d x) a}{b^2 d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 i f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}+\frac {6 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {PolyLog}\left (3,i e^{c+d x}\right )}{b d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3322
Rule 3718
Rule 4180
Rule 4184
Rule 5451
Rule 5567
Rule 5573
Rule 5583
Rule 6589
Rule 6609
Rule 6742
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}-\frac {a \int (e+f x)^3 \text {sech}^2(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {(3 f) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{b d}\\ &=\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {a^2 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {a^2 \int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(3 a f) \int (e+f x)^2 \tanh (c+d x) \, dx}{b^2 d}-\frac {\left (6 i f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b d^2}+\frac {\left (6 i f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b d^2}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {\left (2 a^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2+b^2}+\frac {a^2 \int \left (a (e+f x)^3 \text {sech}^2(c+d x)-b (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(6 a f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{b^2 d}+\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b d^3}-\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b d^3}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}+\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac {a^3 \int (e+f x)^3 \text {sech}^2(c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b \left (a^2+b^2\right )}-\frac {\left (6 a f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 d^2}+\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^4}-\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^4}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}+\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^3 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^3 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}+\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}-\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (3 a^2 f\right ) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{b \left (a^2+b^2\right ) d}-\frac {\left (3 a f^3\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 d^3}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {a^3 (e+f x)^3}{b^2 \left (a^2+b^2\right ) d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 a^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}+\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^3 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^3 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (6 a^3 f\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (6 a^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (6 a^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (6 i a^2 f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}-\frac {\left (6 i a^2 f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}-\frac {\left (3 a f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b^2 d^4}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {a^3 (e+f x)^3}{b^2 \left (a^2+b^2\right ) d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 a^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {3 a^3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b d^4}-\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {3 a f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^3 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^3 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (6 a^3 f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (6 a^2 f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {\left (6 a^2 f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {\left (6 i a^2 f^3\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^3}+\frac {\left (6 i a^2 f^3\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^3}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {a^3 (e+f x)^3}{b^2 \left (a^2+b^2\right ) d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 a^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {3 a^3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {3 a^3 f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b d^4}-\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {3 a f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^3 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^3 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (6 a^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 )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {\left (6 a^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 )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {\left (6 i a^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {\left (6 i a^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {\left (3 a^3 f^3\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^3}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {a^3 (e+f x)^3}{b^2 \left (a^2+b^2\right ) d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 a^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {3 a^3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {3 a^3 f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i a^2 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b d^4}+\frac {6 i a^2 f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {3 a f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^4}+\frac {6 a^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 a^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^3 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^3 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (3 a^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^4}\\ &=-\frac {a (e+f x)^3}{b^2 d}+\frac {a^3 (e+f x)^3}{b^2 \left (a^2+b^2\right ) d}+\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac {6 a^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {3 a^3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {3 a^3 f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i a^2 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b d^4}+\frac {6 i a^2 f^3 \text {Li}_3\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 a^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {3 a f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^4}+\frac {3 a^3 f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^4}+\frac {6 a^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 a^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {(e+f x)^3 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^3 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^3 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 13.31, size = 1118, normalized size = 1.00 \[ \frac {\left (-2 e^3 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^3+f^3 x^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+3 e^2 f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3-f^3 x^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3-3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3-3 e^2 f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+3 f (e+f x)^2 \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d^2-3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d^2-6 e f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d-6 f^3 x \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d+6 e f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+6 f^3 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+6 f^3 \text {Li}_4\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )-6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}+\frac {f \left (-4 a e^{2 c} f^2 x^3 d^3-12 a e e^{2 c} f x^2 d^3-12 a e^2 e^{2 c} x d^3+12 b e^2 \left (1+e^{2 c}\right ) \tan ^{-1}\left (e^{c+d x}\right ) d^2+6 a e^2 \left (1+e^{2 c}\right ) \log \left (1+e^{2 (c+d x)}\right ) d^2+12 i b e \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\text {Li}_2\left (-i e^{c+d x}\right )+\text {Li}_2\left (i e^{c+d x}\right )\right ) d+6 a e \left (1+e^{2 c}\right ) f \left (2 d x \log \left (1+e^{2 (c+d x)}\right )+\text {Li}_2\left (-e^{2 (c+d x)}\right )\right ) d+6 i b \left (1+e^{2 c}\right ) f^2 \left (d^2 \log \left (1-i e^{c+d x}\right ) x^2-d^2 \log \left (1+i e^{c+d x}\right ) x^2-2 d \text {Li}_2\left (-i e^{c+d x}\right ) x+2 d \text {Li}_2\left (i e^{c+d x}\right ) x+2 \text {Li}_3\left (-i e^{c+d x}\right )-2 \text {Li}_3\left (i e^{c+d x}\right )\right )+3 a \left (1+e^{2 c}\right ) f^2 \left (2 d^2 \log \left (1+e^{2 (c+d x)}\right ) x^2+2 d \text {Li}_2\left (-e^{2 (c+d x)}\right ) x-\text {Li}_3\left (-e^{2 (c+d x)}\right )\right )\right )}{2 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right )}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (-b \cosh (c) e^3-a \sinh (d x) e^3-3 b f x \cosh (c) e^2-3 a f x \sinh (d x) e^2-3 b f^2 x^2 \cosh (c) e-3 a f^2 x^2 \sinh (d x) e-b f^3 x^3 \cosh (c)-a f^3 x^3 \sinh (d x)\right )}{\left (a^2+b^2\right ) d} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.78, size = 6525, normalized size = 5.84 \[ \text {result too large to display} \]
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 = 1.74, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\tanh ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]
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 \[ -3 \, a e^{2} f {\left (\frac {2 \, {\left (d x + c\right )}}{{\left (a^{2} + b^{2}\right )} d^{2}} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d^{2}}\right )} + 6 \, b f^{3} \int \frac {x^{2} e^{\left (d x + c\right )}}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} - 6 \, a f^{3} \int \frac {x^{2}}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} + 12 \, b e f^{2} \int \frac {x e^{\left (d x + c\right )}}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} - 12 \, a e f^{2} \int \frac {x}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} + e^{3} {\left (\frac {a^{2} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} d} - \frac {2 \, {\left (b e^{\left (-d x - c\right )} + a\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d}\right )} + \frac {6 \, b e^{2} f \arctan \left (e^{\left (d x + c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d^{2}} + \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} d + b^{2} d + {\left (a^{2} d e^{\left (2 \, c\right )} + b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \int -\frac {2 \, {\left (a^{2} f^{3} x^{3} e^{c} + 3 \, a^{2} e f^{2} x^{2} e^{c} + 3 \, a^{2} e^{2} f x e^{c}\right )} e^{\left (d x\right )}}{a^{2} b + b^{3} - {\left (a^{2} b e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} e^{c} + a b^{2} e^{c}\right )} e^{\left (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 \[ \int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\left (e + f x\right )^{3} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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