Optimal. Leaf size=1049 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.34083, antiderivative size = 1049, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {5589, 5461, 4182, 2531, 6609, 2282, 6589, 5573, 5561, 2190, 6742, 4180, 3718} \[ \frac{6 i b \text{PolyLog}\left (4,-i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac{6 i b \text{PolyLog}\left (4,i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac{6 b^2 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}-\frac{6 b^2 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) f^3}{a \left (a^2+b^2\right ) d^4}+\frac{3 b^2 \text{PolyLog}\left (4,-e^{2 (c+d x)}\right ) f^3}{4 a \left (a^2+b^2\right ) d^4}-\frac{3 \text{PolyLog}\left (4,-e^{2 c+2 d x}\right ) f^3}{4 a d^4}+\frac{3 \text{PolyLog}\left (4,e^{2 c+2 d x}\right ) f^3}{4 a d^4}-\frac{6 i b (e+f x) \text{PolyLog}\left (3,-i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac{6 i b (e+f x) \text{PolyLog}\left (3,i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac{6 b^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}+\frac{6 b^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) f^2}{a \left (a^2+b^2\right ) d^3}-\frac{3 b^2 (e+f x) \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) f^2}{2 a \left (a^2+b^2\right ) d^3}+\frac{3 (e+f x) \text{PolyLog}\left (3,-e^{2 c+2 d x}\right ) f^2}{2 a d^3}-\frac{3 (e+f x) \text{PolyLog}\left (3,e^{2 c+2 d x}\right ) f^2}{2 a d^3}+\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac{3 i b (e+f x)^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac{3 b^2 (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}-\frac{3 b^2 (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) f}{a \left (a^2+b^2\right ) d^2}+\frac{3 b^2 (e+f x)^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) f}{2 a \left (a^2+b^2\right ) d^2}-\frac{3 (e+f x)^2 \text{PolyLog}\left (2,-e^{2 c+2 d x}\right ) f}{2 a d^2}+\frac{3 (e+f x)^2 \text{PolyLog}\left (2,e^{2 c+2 d x}\right ) f}{2 a d^2}-\frac{2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5589
Rule 5461
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 5573
Rule 5561
Rule 2190
Rule 6742
Rule 4180
Rule 3718
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \text{csch}(c+d x) \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \text{csch}(c+d x) \text{sech}(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac{2 \int (e+f x)^3 \text{csch}(2 c+2 d x) \, dx}{a}-\frac{b \int (e+f x)^3 \text{sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int \frac{(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac{b^2 (e+f x)^4}{4 a \left (a^2+b^2\right ) f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b \int \left (a (e+f x)^3 \text{sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int \frac{e^{c+d x} (e+f x)^3}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int \frac{e^{c+d x} (e+f x)^3}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\frac{(3 f) \int (e+f x)^2 \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}+\frac{(3 f) \int (e+f x)^2 \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}\\ &=\frac{b^2 (e+f x)^4}{4 a \left (a^2+b^2\right ) f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac{b \int (e+f x)^3 \text{sech}(c+d x) \, dx}{a^2+b^2}+\frac{b^2 \int (e+f x)^3 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}+\frac{\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac{\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac{\left (3 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a d^2}-\frac{\left (3 f^2\right ) \int (e+f x) \text{Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a d^2}\\ &=-\frac{2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}+\frac{\left (2 b^2\right ) \int \frac{e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )}+\frac{(3 i b f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac{(3 i b f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac{\left (6 b^2 f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}+\frac{\left (6 b^2 f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}-\frac{\left (3 f^3\right ) \int \text{Li}_3\left (-e^{2 c+2 d x}\right ) \, dx}{2 a d^3}+\frac{\left (3 f^3\right ) \int \text{Li}_3\left (e^{2 c+2 d x}\right ) \, dx}{2 a d^3}\\ &=-\frac{2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac{\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d}-\frac{\left (6 i b f^2\right ) \int (e+f x) \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac{\left (6 i b f^2\right ) \int (e+f x) \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a d^4}+\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a d^4}-\frac{\left (6 b^2 f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}-\frac{\left (6 b^2 f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}\\ &=-\frac{2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac{6 i b f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 i b f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac{3 f^3 \text{Li}_4\left (-e^{2 c+2 d x}\right )}{4 a d^4}+\frac{3 f^3 \text{Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4}-\frac{\left (3 b^2 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}-\frac{\left (6 b^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac{\left (6 b^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac{\left (6 i b f^3\right ) \int \text{Li}_3\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3}-\frac{\left (6 i b f^3\right ) \int \text{Li}_3\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3}\\ &=-\frac{2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac{6 i b f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 i b f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac{3 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac{6 b^2 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac{3 f^3 \text{Li}_4\left (-e^{2 c+2 d x}\right )}{4 a d^4}+\frac{3 f^3 \text{Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4}+\frac{\left (6 i b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac{\left (6 i b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac{\left (3 b^2 f^3\right ) \int \text{Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 a \left (a^2+b^2\right ) d^3}\\ &=-\frac{2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac{6 i b f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 i b f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac{3 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}+\frac{6 i b f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac{6 i b f^3 \text{Li}_4\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac{3 f^3 \text{Li}_4\left (-e^{2 c+2 d x}\right )}{4 a d^4}+\frac{3 f^3 \text{Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4}+\frac{\left (3 b^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a \left (a^2+b^2\right ) d^4}\\ &=-\frac{2 b (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 i b f (e+f x)^2 \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 i b f (e+f x)^2 \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac{6 i b f^2 (e+f x) \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 i b f^2 (e+f x) \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac{3 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}+\frac{6 i b f^3 \text{Li}_4\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac{6 i b f^3 \text{Li}_4\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^4}+\frac{3 b^2 f^3 \text{Li}_4\left (-e^{2 (c+d x)}\right )}{4 a \left (a^2+b^2\right ) d^4}-\frac{3 f^3 \text{Li}_4\left (-e^{2 c+2 d x}\right )}{4 a d^4}+\frac{3 f^3 \text{Li}_4\left (e^{2 c+2 d x}\right )}{4 a d^4}\\ \end{align*}
Mathematica [B] time = 32.8475, size = 3000, normalized size = 2.86 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.756, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}{\rm csch} \left (dx+c\right ){\rm sech} \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 = 3.69805, size = 5839, normalized size = 5.57 \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(-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} \operatorname{csch}\left (d x + c\right ) \operatorname{sech}\left (d x + c\right )}{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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