Optimal. Leaf size=1245 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 3.43733, antiderivative size = 1245, normalized size of antiderivative = 1., number of steps used = 88, number of rules used = 33, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {5589, 2622, 288, 321, 207, 5462, 6688, 12, 6742, 6273, 4182, 2531, 2282, 6589, 4133, 453, 203, 4180, 2279, 2391, 2621, 5203, 3770, 5461, 4184, 3716, 2190, 6741, 5573, 3322, 2264, 3718, 5451} \[ -\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^3}+\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2 \text{sech}(c+d x) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 b^3}{a^2 \left (a^2+b^2\right ) d}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}+\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2 \tanh (c+d x) b^3}{a^2 \left (a^2+b^2\right ) d}-\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a^3 d^2}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right ) b^2}{a^3 d}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right ) b^2}{a^3 d^2}+\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{a^3 d^3}-\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) b^2}{a^3 d^3}+\frac{2 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right ) b^2}{a^3 d^2}+\frac{2 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right ) b^2}{a^3 d^3}-\frac{2 f^2 \text{PolyLog}\left (3,e^{c+d x}\right ) b^2}{a^3 d^3}+\frac{(e+f x)^2 \text{sech}(c+d x) b^2}{a^3 d}+\frac{2 (e+f x)^2 b}{a^2 d}+\frac{2 (e+f x)^2 \coth (2 c+2 d x) b}{a^2 d}-\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right ) b}{a^2 d^2}-\frac{f^2 \text{PolyLog}\left (2,e^{4 (c+d x)}\right ) b}{2 a^2 d^3}+\frac{4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}+\frac{3 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{3 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5589
Rule 2622
Rule 288
Rule 321
Rule 207
Rule 5462
Rule 6688
Rule 12
Rule 6742
Rule 6273
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 4133
Rule 453
Rule 203
Rule 4180
Rule 2279
Rule 2391
Rule 2621
Rule 5203
Rule 3770
Rule 5461
Rule 4184
Rule 3716
Rule 2190
Rule 6741
Rule 5573
Rule 3322
Rule 2264
Rule 3718
Rule 5451
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \text{csch}^3(c+d x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \text{csch}^3(c+d x) \text{sech}^2(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac{3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b \int (e+f x)^2 \text{csch}^2(c+d x) \text{sech}^2(c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x)^2 \text{csch}(c+d x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac{(2 f) \int (e+f x) \left (\frac{3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{3 \text{sech}(c+d x)}{2 d}-\frac{\text{csch}^2(c+d x) \text{sech}(c+d x)}{2 d}\right ) \, dx}{a}\\ &=\frac{3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{(4 b) \int (e+f x)^2 \text{csch}^2(2 c+2 d x) \, dx}{a^2}+\frac{b^2 \int (e+f x)^2 \text{csch}(c+d x) \text{sech}^2(c+d x) \, dx}{a^3}-\frac{b^3 \int \frac{(e+f x)^2 \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac{(2 f) \int \frac{(e+f x) \left (3 \tanh ^{-1}(\cosh (c+d x))-\left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x)\right )}{2 d} \, dx}{a}\\ &=\frac{3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 \int (e+f x)^2 \text{sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac{b^5 \int \frac{(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac{\left (2 b^2 f\right ) \int (e+f x) \left (-\frac{\tanh ^{-1}(\cosh (c+d x))}{d}+\frac{\text{sech}(c+d x)}{d}\right ) \, dx}{a^3}-\frac{f \int (e+f x) \left (3 \tanh ^{-1}(\cosh (c+d x))-\left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x)\right ) \, dx}{a d}-\frac{(4 b f) \int (e+f x) \coth (2 c+2 d x) \, dx}{a^2 d}\\ &=\frac{2 b (e+f x)^2}{a^2 d}+\frac{3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 \int \left (a (e+f x)^2 \text{sech}^2(c+d x)-b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a^3 \left (a^2+b^2\right )}-\frac{\left (2 b^5\right ) \int \frac{e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac{\left (2 b^2 f\right ) \int \frac{(e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text{sech}(c+d x)\right )}{d} \, dx}{a^3}-\frac{f \int \left (3 (e+f x) \tanh ^{-1}(\cosh (c+d x))-(e+f x) \left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x)\right ) \, dx}{a d}+\frac{(8 b f) \int \frac{e^{2 (2 c+2 d x)} (e+f x)}{1-e^{2 (2 c+2 d x)}} \, dx}{a^2 d}\\ &=\frac{2 b (e+f x)^2}{a^2 d}+\frac{3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{\left (2 b^6\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^{3/2}}+\frac{\left (2 b^6\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^{3/2}}-\frac{b^3 \int (e+f x)^2 \text{sech}^2(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac{b^4 \int (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac{f \int (e+f x) \left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x) \, dx}{a d}-\frac{(3 f) \int (e+f x) \tanh ^{-1}(\cosh (c+d x)) \, dx}{a d}-\frac{\left (2 b^2 f\right ) \int (e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text{sech}(c+d x)\right ) \, dx}{a^3 d}+\frac{\left (2 b f^2\right ) \int \log \left (1-e^{2 (2 c+2 d x)}\right ) \, dx}{a^2 d^2}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac{3 \int d (e+f x)^2 \text{csch}(c+d x) \, dx}{2 a d}+\frac{f \int \left (e \left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x)+f x \left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x)\right ) \, dx}{a d}-\frac{\left (2 b^2 f\right ) \int \left (-(e+f x) \tanh ^{-1}(\cosh (c+d x))+(e+f x) \text{sech}(c+d x)\right ) \, dx}{a^3 d}+\frac{\left (2 b^5 f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d}-\frac{\left (2 b^5 f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{\left (2 b^3 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac{\left (2 b^4 f\right ) \int (e+f x) \text{sech}(c+d x) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (2 c+2 d x)}\right )}{2 a^2 d^3}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac{3 \int (e+f x)^2 \text{csch}(c+d x) \, dx}{2 a}+\frac{\left (2 b^2 f\right ) \int (e+f x) \tanh ^{-1}(\cosh (c+d x)) \, dx}{a^3 d}-\frac{\left (2 b^2 f\right ) \int (e+f x) \text{sech}(c+d x) \, dx}{a^3 d}+\frac{\left (4 b^3 f\right ) \int \frac{e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac{(e f) \int \left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x) \, dx}{a d}+\frac{\left (2 b^5 f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac{\left (2 b^5 f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac{\left (2 i b^4 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac{\left (2 i b^4 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac{f^2 \int x \left (3+\text{csch}^2(c+d x)\right ) \text{sech}(c+d x) \, dx}{a d}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \int d (e+f x)^2 \text{csch}(c+d x) \, dx}{a^3 d}+\frac{(3 f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac{(3 f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac{(e f) \operatorname{Subst}\left (\int \frac{-1-3 x^2}{x^2 \left (1+x^2\right )} \, dx,x,\sinh (c+d x)\right )}{a d^2}+\frac{\left (2 b^5 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{\left (2 b^5 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{\left (2 i b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{\left (2 i b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{\left (2 i b^2 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac{\left (2 i b^2 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac{\left (2 b^3 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}+\frac{f^2 \int \left (3 x \text{sech}(c+d x)+x \text{csch}^2(c+d x) \text{sech}(c+d x)\right ) \, dx}{a d}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 \int (e+f x)^2 \text{csch}(c+d x) \, dx}{a^3}+\frac{(2 e f) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{a d^2}+\frac{\left (2 i b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac{\left (2 i b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac{\left (b^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{\left (3 f^2\right ) \int \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (3 f^2\right ) \int \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}+\frac{f^2 \int x \text{csch}^2(c+d x) \text{sech}(c+d x) \, dx}{a d}+\frac{\left (3 f^2\right ) \int x \text{sech}(c+d x) \, dx}{a d}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac{f^2 x \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{2 i b^2 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{2 i b^2 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac{\left (2 b^2 f\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac{\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}-\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac{\left (3 i f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (3 i f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}-\frac{f^2 \int \left (-\frac{\tan ^{-1}(\sinh (c+d x))}{d}-\frac{\text{csch}(c+d x)}{d}\right ) \, dx}{a d}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac{f^2 x \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac{2 i b^2 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{2 i b^2 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{2 b^2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac{\left (3 i f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac{\left (3 i f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac{f^2 \int \tan ^{-1}(\sinh (c+d x)) \, dx}{a d^2}+\frac{f^2 \int \text{csch}(c+d x) \, dx}{a d^2}+\frac{\left (2 b^2 f^2\right ) \int \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac{\left (2 b^2 f^2\right ) \int \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a^3 d^2}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac{3 i f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{2 i b^2 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{3 i f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 i b^2 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{2 b^2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{\left (2 b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac{\left (2 b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac{f^2 \int d x \text{sech}(c+d x) \, dx}{a d^2}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac{3 i f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{2 i b^2 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{3 i f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 i b^2 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{2 b^2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{2 b^2 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac{f^2 \int x \text{sech}(c+d x) \, dx}{a d}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac{3 i f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{2 i b^2 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{3 i f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 i b^2 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{2 b^2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{2 b^2 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{\left (i f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}-\frac{\left (i f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac{3 i f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{2 i b^2 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{3 i f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 i b^2 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{2 b^2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{2 b^2 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{\left (i f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac{\left (i f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=\frac{2 b (e+f x)^2}{a^2 d}-\frac{b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac{4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac{4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac{2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac{e f \text{csch}(c+d x)}{a d^2}-\frac{f^2 x \text{csch}(c+d x)}{a d^2}-\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{b^5 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac{2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac{2 i f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac{2 i b^2 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac{2 i b^4 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac{2 i f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac{2 i b^2 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac{2 i b^4 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{2 b^2 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 b^5 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{b f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{2 b^2 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 b^5 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac{3 (e+f x)^2 \text{sech}(c+d x)}{2 a d}+\frac{b^2 (e+f x)^2 \text{sech}(c+d x)}{a^3 d}-\frac{b^4 (e+f x)^2 \text{sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{csch}^2(c+d x) \text{sech}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [B] time = 27.5585, size = 2574, normalized size = 2.07 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.557, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ({\rm csch} \left (dx+c\right ) \right ) ^{3} \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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(-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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