Optimal. Leaf size=1443 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.18695, antiderivative size = 1443, normalized size of antiderivative = 1., number of steps used = 55, number of rules used = 18, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5579, 5448, 3296, 2638, 5447, 3311, 32, 2635, 8, 3310, 5565, 5446, 5561, 2190, 2531, 6609, 2282, 6589} \[ -\frac{6 f^3 \cosh (c+d x) a^4}{b^5 d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x) a^4}{b^5 d^2}+\frac{(e+f x)^3 \sinh (c+d x) a^4}{b^5 d}+\frac{6 f^2 (e+f x) \sinh (c+d x) a^4}{b^5 d^3}+\frac{\left (a^2+b^2\right ) (e+f x)^4 a^3}{4 b^6 f}-\frac{(e+f x)^3 a^3}{4 b^4 d}-\frac{(e+f x)^3 \sinh ^2(c+d x) a^3}{2 b^4 d}-\frac{3 f^2 (e+f x) \sinh ^2(c+d x) a^3}{4 b^4 d^3}-\frac{3 f^3 x a^3}{8 b^4 d^3}-\frac{\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac{\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^3}{b^6 d}-\frac{3 \left (a^2+b^2\right ) f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^2}-\frac{3 \left (a^2+b^2\right ) f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^2}+\frac{6 \left (a^2+b^2\right ) f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^3}+\frac{6 \left (a^2+b^2\right ) f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^3}-\frac{6 \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^4}-\frac{6 \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{b^6 d^4}+\frac{3 f^3 \cosh (c+d x) \sinh (c+d x) a^3}{8 b^4 d^4}+\frac{3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x) a^3}{4 b^4 d^2}-\frac{2 f^3 \cosh ^3(c+d x) a^2}{27 b^3 d^4}-\frac{f (e+f x)^2 \cosh ^3(c+d x) a^2}{3 b^3 d^2}-\frac{40 f^3 \cosh (c+d x) a^2}{9 b^3 d^4}-\frac{2 f (e+f x)^2 \cosh (c+d x) a^2}{b^3 d^2}+\frac{2 (e+f x)^3 \sinh (c+d x) a^2}{3 b^3 d}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x) a^2}{3 b^3 d}+\frac{2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x) a^2}{9 b^3 d^3}+\frac{40 f^2 (e+f x) \sinh (c+d x) a^2}{9 b^3 d^3}-\frac{(e+f x)^3 \cosh ^4(c+d x) a}{4 b^2 d}-\frac{3 f^2 (e+f x) \cosh ^4(c+d x) a}{32 b^2 d^3}+\frac{3 (e+f x)^3 a}{32 b^2 d}-\frac{9 f^2 (e+f x) \cosh ^2(c+d x) a}{32 b^2 d^3}+\frac{45 f^3 x a}{256 b^2 d^3}+\frac{3 f^3 \cosh ^3(c+d x) \sinh (c+d x) a}{128 b^2 d^4}+\frac{3 f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x) a}{16 b^2 d^2}+\frac{45 f^3 \cosh (c+d x) \sinh (c+d x) a}{256 b^2 d^4}+\frac{9 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x) a}{32 b^2 d^2}+\frac{3 f^3 \cosh (c+d x)}{4 b d^4}+\frac{3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac{f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac{f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac{3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac{3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}-\frac{3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}+\frac{3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5579
Rule 5448
Rule 3296
Rule 2638
Rule 5447
Rule 3311
Rule 32
Rule 2635
Rule 8
Rule 3310
Rule 5565
Rule 5446
Rule 5561
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^3 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{a \int (e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac{\int \left (-\frac{1}{8} (e+f x)^3 \cosh (c+d x)+\frac{1}{16} (e+f x)^3 \cosh (3 c+3 d x)+\frac{1}{16} (e+f x)^3 \cosh (5 c+5 d x)\right ) \, dx}{b}\\ &=-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}+\frac{a^2 \int (e+f x)^3 \cosh ^3(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac{\int (e+f x)^3 \cosh (3 c+3 d x) \, dx}{16 b}+\frac{\int (e+f x)^3 \cosh (5 c+5 d x) \, dx}{16 b}-\frac{\int (e+f x)^3 \cosh (c+d x) \, dx}{8 b}+\frac{(3 a f) \int (e+f x)^2 \cosh ^4(c+d x) \, dx}{4 b^2 d}\\ &=-\frac{a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac{3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac{a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}+\frac{a^4 \int (e+f x)^3 \cosh (c+d x) \, dx}{b^5}-\frac{a^3 \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{b^4}+\frac{\left (2 a^2\right ) \int (e+f x)^3 \cosh (c+d x) \, dx}{3 b^3}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac{(9 a f) \int (e+f x)^2 \cosh ^2(c+d x) \, dx}{16 b^2 d}-\frac{(3 f) \int (e+f x)^2 \sinh (5 c+5 d x) \, dx}{80 b d}-\frac{f \int (e+f x)^2 \sinh (3 c+3 d x) \, dx}{16 b d}+\frac{(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{8 b d}+\frac{\left (2 a^2 f^2\right ) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 b^3 d^2}+\frac{\left (3 a f^3\right ) \int \cosh ^4(c+d x) \, dx}{32 b^2 d^3}\\ &=\frac{a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}+\frac{3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac{9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac{a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac{3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac{3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}+\frac{a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac{9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac{3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^3}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac{\left (a^3 \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^3}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac{\left (3 a^4 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^5 d}+\frac{\left (3 a^3 f\right ) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 b^4 d}-\frac{\left (2 a^2 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^3 d}+\frac{(9 a f) \int (e+f x)^2 \, dx}{32 b^2 d}+\frac{\left (4 a^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{9 b^3 d^2}+\frac{\left (3 f^2\right ) \int (e+f x) \cosh (5 c+5 d x) \, dx}{200 b d^2}+\frac{f^2 \int (e+f x) \cosh (3 c+3 d x) \, dx}{24 b d^2}-\frac{\left (3 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{4 b d^2}+\frac{\left (9 a f^3\right ) \int \cosh ^2(c+d x) \, dx}{128 b^2 d^3}+\frac{\left (9 a f^3\right ) \int \cosh ^2(c+d x) \, dx}{32 b^2 d^3}\\ &=\frac{3 a (e+f x)^3}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac{3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac{3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac{9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac{a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac{3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac{3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}+\frac{4 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac{3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac{45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac{3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac{f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac{\left (3 a^3 f\right ) \int (e+f x)^2 \, dx}{4 b^4 d}+\frac{\left (3 a^3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d}+\frac{\left (3 a^3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d}+\frac{\left (6 a^4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^5 d^2}+\frac{\left (4 a^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^3 d^2}+\frac{\left (3 a^3 f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 b^4 d^3}-\frac{\left (4 a^2 f^3\right ) \int \sinh (c+d x) \, dx}{9 b^3 d^3}+\frac{\left (9 a f^3\right ) \int 1 \, dx}{256 b^2 d^3}+\frac{\left (9 a f^3\right ) \int 1 \, dx}{64 b^2 d^3}-\frac{\left (3 f^3\right ) \int \sinh (5 c+5 d x) \, dx}{1000 b d^3}-\frac{f^3 \int \sinh (3 c+3 d x) \, dx}{72 b d^3}+\frac{\left (3 f^3\right ) \int \sinh (c+d x) \, dx}{4 b d^3}\\ &=\frac{45 a f^3 x}{256 b^2 d^3}-\frac{a^3 (e+f x)^3}{4 b^4 d}+\frac{3 a (e+f x)^3}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac{4 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac{3 f^3 \cosh (c+d x)}{4 b d^4}-\frac{3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac{3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac{9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac{a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac{3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac{f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac{3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac{3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^2}+\frac{6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac{40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac{3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac{3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac{45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac{3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac{f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}+\frac{\left (6 a^3 \left (a^2+b^2\right ) 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}{b^6 d^2}+\frac{\left (6 a^3 \left (a^2+b^2\right ) 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}{b^6 d^2}-\frac{\left (6 a^4 f^3\right ) \int \sinh (c+d x) \, dx}{b^5 d^3}-\frac{\left (3 a^3 f^3\right ) \int 1 \, dx}{8 b^4 d^3}-\frac{\left (4 a^2 f^3\right ) \int \sinh (c+d x) \, dx}{b^3 d^3}\\ &=-\frac{3 a^3 f^3 x}{8 b^4 d^3}+\frac{45 a f^3 x}{256 b^2 d^3}-\frac{a^3 (e+f x)^3}{4 b^4 d}+\frac{3 a (e+f x)^3}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac{6 a^4 f^3 \cosh (c+d x)}{b^5 d^4}-\frac{40 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac{3 f^3 \cosh (c+d x)}{4 b d^4}-\frac{3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac{3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac{9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac{a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac{3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac{f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac{3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac{3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^2}+\frac{6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^3}+\frac{6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^3}+\frac{6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac{40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac{3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac{3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac{45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac{3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac{f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac{\left (6 a^3 \left (a^2+b^2\right ) f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d^3}-\frac{\left (6 a^3 \left (a^2+b^2\right ) f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^6 d^3}\\ &=-\frac{3 a^3 f^3 x}{8 b^4 d^3}+\frac{45 a f^3 x}{256 b^2 d^3}-\frac{a^3 (e+f x)^3}{4 b^4 d}+\frac{3 a (e+f x)^3}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac{6 a^4 f^3 \cosh (c+d x)}{b^5 d^4}-\frac{40 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac{3 f^3 \cosh (c+d x)}{4 b d^4}-\frac{3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac{3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac{9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac{a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac{3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac{f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac{3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac{3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^2}+\frac{6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^3}+\frac{6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^3}+\frac{6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac{40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac{3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac{3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac{45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac{3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac{f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}-\frac{\left (6 a^3 \left (a^2+b^2\right ) 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 )}{b^6 d^4}-\frac{\left (6 a^3 \left (a^2+b^2\right ) 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 )}{b^6 d^4}\\ &=-\frac{3 a^3 f^3 x}{8 b^4 d^3}+\frac{45 a f^3 x}{256 b^2 d^3}-\frac{a^3 (e+f x)^3}{4 b^4 d}+\frac{3 a (e+f x)^3}{32 b^2 d}+\frac{a^3 \left (a^2+b^2\right ) (e+f x)^4}{4 b^6 f}-\frac{6 a^4 f^3 \cosh (c+d x)}{b^5 d^4}-\frac{40 a^2 f^3 \cosh (c+d x)}{9 b^3 d^4}+\frac{3 f^3 \cosh (c+d x)}{4 b d^4}-\frac{3 a^4 f (e+f x)^2 \cosh (c+d x)}{b^5 d^2}-\frac{2 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac{3 f (e+f x)^2 \cosh (c+d x)}{8 b d^2}-\frac{9 a f^2 (e+f x) \cosh ^2(c+d x)}{32 b^2 d^3}-\frac{2 a^2 f^3 \cosh ^3(c+d x)}{27 b^3 d^4}-\frac{a^2 f (e+f x)^2 \cosh ^3(c+d x)}{3 b^3 d^2}-\frac{3 a f^2 (e+f x) \cosh ^4(c+d x)}{32 b^2 d^3}-\frac{a (e+f x)^3 \cosh ^4(c+d x)}{4 b^2 d}-\frac{f^3 \cosh (3 c+3 d x)}{216 b d^4}-\frac{f (e+f x)^2 \cosh (3 c+3 d x)}{48 b d^2}-\frac{3 f^3 \cosh (5 c+5 d x)}{5000 b d^4}-\frac{3 f (e+f x)^2 \cosh (5 c+5 d x)}{400 b d^2}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{a^3 \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^2}-\frac{3 a^3 \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^2}+\frac{6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^3}+\frac{6 a^3 \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^3}-\frac{6 a^3 \left (a^2+b^2\right ) f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^6 d^4}-\frac{6 a^3 \left (a^2+b^2\right ) f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^6 d^4}+\frac{6 a^4 f^2 (e+f x) \sinh (c+d x)}{b^5 d^3}+\frac{40 a^2 f^2 (e+f x) \sinh (c+d x)}{9 b^3 d^3}-\frac{3 f^2 (e+f x) \sinh (c+d x)}{4 b d^3}+\frac{a^4 (e+f x)^3 \sinh (c+d x)}{b^5 d}+\frac{2 a^2 (e+f x)^3 \sinh (c+d x)}{3 b^3 d}-\frac{(e+f x)^3 \sinh (c+d x)}{8 b d}+\frac{3 a^3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^4 d^4}+\frac{45 a f^3 \cosh (c+d x) \sinh (c+d x)}{256 b^2 d^4}+\frac{3 a^3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac{9 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac{2 a^2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 b^3 d^3}+\frac{a^2 (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac{3 a f^3 \cosh ^3(c+d x) \sinh (c+d x)}{128 b^2 d^4}+\frac{3 a f (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac{3 a^3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b^4 d^3}-\frac{a^3 (e+f x)^3 \sinh ^2(c+d x)}{2 b^4 d}+\frac{f^2 (e+f x) \sinh (3 c+3 d x)}{72 b d^3}+\frac{(e+f x)^3 \sinh (3 c+3 d x)}{48 b d}+\frac{3 f^2 (e+f x) \sinh (5 c+5 d x)}{1000 b d^3}+\frac{(e+f x)^3 \sinh (5 c+5 d x)}{80 b d}\\ \end{align*}
Mathematica [B] time = 21.8732, size = 5157, normalized size = 3.57 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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