Optimal. Leaf size=1118 \[ -\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 \text {ArcTan}\left (e^{c+d x}\right )}{b d^2}-\frac {6 a^2 f (e+f x)^2 \text {ArcTan}\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 {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {6 i a^2 f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{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 )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {PolyLog}\left (2,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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {3 a^3 f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^4}-\frac {6 i a^2 f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \text {PolyLog}\left (3,i e^{c+d x}\right )}{b d^4}+\frac {6 i a^2 f^3 \text {PolyLog}\left (3,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 {PolyLog}\left (3,-\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 {PolyLog}\left (3,-\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 {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b^2 d^4}+\frac {3 a^3 f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^4}+\frac {6 a^2 f^3 \text {PolyLog}\left (4,-\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 {PolyLog}\left (4,-\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} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.62, 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 = {5686,
5559, 4265, 2611, 2320, 6724, 5702, 4269, 3799, 2221, 5692, 3403, 2296, 6744, 6874}
\begin {gather*} \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 \text {ArcTan}\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 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3403
Rule 3799
Rule 4265
Rule 4269
Rule 5559
Rule 5686
Rule 5692
Rule 5702
Rule 6724
Rule 6744
Rule 6874
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 13.55, size = 1614, normalized size = 1.44 \begin {gather*} \frac {f \left (-12 a d^3 e^2 e^{2 c} x+12 a d^3 e^2 \left (1+e^{2 c}\right ) x+12 a d^3 e f x^2+4 a d^3 f^2 x^3+12 b d^2 e^2 \left (1+e^{2 c}\right ) \text {ArcTan}\left (e^{c+d x}\right )-6 a d^2 e^2 \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+12 i b d 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 {PolyLog}\left (2,-i e^{c+d x}\right )+\text {PolyLog}\left (2,i e^{c+d x}\right )\right )-6 a d e \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )+6 i b \left (1+e^{2 c}\right ) f^2 \left (d^2 x^2 \log \left (1-i e^{c+d x}\right )-d^2 x^2 \log \left (1+i e^{c+d x}\right )-2 d x \text {PolyLog}\left (2,-i e^{c+d x}\right )+2 d x \text {PolyLog}\left (2,i e^{c+d x}\right )+2 \text {PolyLog}\left (3,-i e^{c+d x}\right )-2 \text {PolyLog}\left (3,i e^{c+d x}\right )\right )-a \left (1+e^{2 c}\right ) f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1+e^{2 (c+d x)}\right )\right )-6 d x \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )+3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )\right )}{2 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right )}-\frac {a^2 \left (2 d^3 e^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}} \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )+3 \sqrt {-a^2-b^2} d^3 e^2 e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {-a^2-b^2} d^3 e e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\sqrt {-a^2-b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {-a^2-b^2} d^3 e^2 e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {-a^2-b^2} d^3 e e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\sqrt {-a^2-b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {-a^2-b^2} d^2 e^c f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {-a^2-b^2} d^2 e^c f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {-a^2-b^2} d e e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {-a^2-b^2} d e^c f^3 x \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {-a^2-b^2} d e e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {-a^2-b^2} d e^c f^3 x \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {-a^2-b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {-a^2-b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{\left (-a^2-b^2\right )^{3/2} d^4 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (-b e^3 \cosh (c)-3 b e^2 f x \cosh (c)-3 b e f^2 x^2 \cosh (c)-b f^3 x^3 \cosh (c)-a e^3 \sinh (d x)-3 a e^2 f x \sinh (d x)-3 a e f^2 x^2 \sinh (d x)-a f^3 x^3 \sinh (d x)\right )}{\left (a^2+b^2\right ) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.06, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 10804 vs. \(2 (1046) = 2092\).
time = 0.56, size = 10804, normalized size = 9.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{3} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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