Optimal. Leaf size=1519 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.15352, antiderivative size = 1519, normalized size of antiderivative = 1., number of steps used = 61, number of rules used = 15, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.441, Rules used = {5581, 5449, 3296, 2638, 4180, 2531, 6609, 2282, 6589, 3718, 2190, 5567, 5573, 5561, 6742} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5581
Rule 5449
Rule 3296
Rule 2638
Rule 4180
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 3718
Rule 2190
Rule 5567
Rule 5573
Rule 5561
Rule 6742
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \sinh (c+d x) \tanh (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{a \int (e+f x)^3 \tanh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac{\int (e+f x)^3 \cosh (c+d x) \, dx}{b}-\frac{\int (e+f x)^3 \text{sech}(c+d x) \, dx}{b}\\ &=\frac{a (e+f x)^4}{4 b^2 f}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}+\frac{a^2 \int (e+f x)^3 \text{sech}(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^3 \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac{(2 a) \int \frac{e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{b^2}+\frac{(3 i f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{b d}-\frac{(3 i f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{b d}-\frac{(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{b d}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}-\frac{a^3 \int (e+f x)^3 \text{sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{b^3 \left (a^2+b^2\right )}-\frac{a^3 \int \frac{(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac{\left (3 i a^2 f\right ) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{b^3 d}+\frac{\left (3 i a^2 f\right ) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{b^3 d}+\frac{(3 a f) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 d}-\frac{\left (6 i f^2\right ) \int (e+f x) \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{b d^2}+\frac{\left (6 i f^2\right ) \int (e+f x) \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{b d^2}+\frac{\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b d^2}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{a^3 (e+f x)^4}{4 b^2 \left (a^2+b^2\right ) f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}-\frac{a^3 \int \left (a (e+f x)^3 \text{sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{b^3 \left (a^2+b^2\right )}-\frac{a^3 \int \frac{e^{c+d x} (e+f x)^3}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b \left (a^2+b^2\right )}-\frac{a^3 \int \frac{e^{c+d x} (e+f x)^3}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{b \left (a^2+b^2\right )}+\frac{\left (6 i a^2 f^2\right ) \int (e+f x) \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^3 d^2}-\frac{\left (6 i a^2 f^2\right ) \int (e+f x) \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{b^3 d^2}+\frac{\left (3 a f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 d^2}+\frac{\left (6 i f^3\right ) \int \text{Li}_3\left (-i e^{c+d x}\right ) \, dx}{b d^3}-\frac{\left (6 i f^3\right ) \int \text{Li}_3\left (i e^{c+d x}\right ) \, dx}{b d^3}-\frac{\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{b d^3}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{a^3 (e+f x)^4}{4 b^2 \left (a^2+b^2\right ) f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{6 f^3 \cosh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{3 a f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}-\frac{a^4 \int (e+f x)^3 \text{sech}(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac{a^3 \int (e+f x)^3 \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )}+\frac{\left (3 a^3 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^2 \left (a^2+b^2\right ) d}+\frac{\left (3 a^3 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^2 \left (a^2+b^2\right ) d}+\frac{\left (6 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-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}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^4}-\frac{\left (6 i a^2 f^3\right ) \int \text{Li}_3\left (-i e^{c+d x}\right ) \, dx}{b^3 d^3}+\frac{\left (6 i a^2 f^3\right ) \int \text{Li}_3\left (i e^{c+d x}\right ) \, dx}{b^3 d^3}-\frac{\left (3 a f^3\right ) \int \text{Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 b^2 d^3}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac{6 f^3 \cosh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{3 a f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac{6 i f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b d^4}-\frac{6 i f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b d^4}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}+\frac{\left (2 a^3\right ) \int \frac{e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )}+\frac{\left (3 i a^4 f\right ) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d}-\frac{\left (3 i a^4 f\right ) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (6 a^3 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^2 \left (a^2+b^2\right ) d^2}+\frac{\left (6 a^3 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^2 \left (a^2+b^2\right ) d^2}-\frac{\left (6 i a^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^4}+\frac{\left (6 i a^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^4}-\frac{\left (3 a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 b^2 d^4}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac{6 f^3 \cosh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac{a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{3 a f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac{6 i a^2 f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac{6 i f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac{6 i a^2 f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac{6 i f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac{3 a f^3 \text{Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}-\frac{\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac{\left (6 i a^4 f^2\right ) \int (e+f x) \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^2}+\frac{\left (6 i a^4 f^2\right ) \int (e+f x) \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^2}-\frac{\left (6 a^3 f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^3}-\frac{\left (6 a^3 f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^3}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac{6 f^3 \cosh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac{a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{3 a f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac{6 i a^2 f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac{6 i f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac{6 i a^2 f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac{6 i f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac{3 a f^3 \text{Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}-\frac{\left (3 a^3 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}-\frac{\left (6 a^3 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^2 \left (a^2+b^2\right ) d^4}-\frac{\left (6 a^3 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^2 \left (a^2+b^2\right ) d^4}+\frac{\left (6 i a^4 f^3\right ) \int \text{Li}_3\left (-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^3}-\frac{\left (6 i a^4 f^3\right ) \int \text{Li}_3\left (i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^3}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac{6 f^3 \cosh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac{a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{3 a f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac{3 a^3 f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}-\frac{6 i a^2 f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac{6 i f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac{6 i a^2 f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac{6 i f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac{6 a^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac{6 a^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac{3 a f^3 \text{Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}+\frac{\left (6 i a^4 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}-\frac{\left (6 i a^4 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}+\frac{\left (3 a^3 f^3\right ) \int \text{Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 b^2 \left (a^2+b^2\right ) d^3}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac{6 f^3 \cosh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac{a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{3 a f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac{3 a^3 f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}-\frac{6 i a^2 f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac{6 i f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac{6 i a^4 f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}+\frac{6 i a^2 f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac{6 i f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac{6 i a^4 f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}-\frac{6 a^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac{6 a^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac{3 a f^3 \text{Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}+\frac{\left (3 a^3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right ) d^4}\\ &=\frac{a (e+f x)^4}{4 b^2 f}+\frac{2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac{2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac{6 f^3 \cosh (c+d x)}{b d^4}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac{a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{3 i a^2 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{3 i a^4 f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac{3 a^3 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac{6 i f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac{6 i a^2 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac{6 i f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{6 i a^4 f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{6 a^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{3 a f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac{3 a^3 f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}-\frac{6 i a^2 f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac{6 i f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac{6 i a^4 f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}+\frac{6 i a^2 f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac{6 i f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac{6 i a^4 f^3 \text{Li}_4\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}-\frac{6 a^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac{6 a^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac{3 a f^3 \text{Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac{3 a^3 f^3 \text{Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right ) d^4}+\frac{6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac{(e+f x)^3 \sinh (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 26.2135, size = 2861, normalized size = 1.88 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}\tanh \left ( dx+c \right ) }{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 [C] time = 4.30697, size = 10368, normalized size = 6.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e + f x\right )^{3} \sinh ^{2}{\left (c + d x \right )} \tanh{\left (c + d x \right )}}{a + b \sinh{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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