Optimal. Leaf size=904 \[ -\frac{(e+f x)^2 a^4}{b^3 \left (a^2+b^2\right ) d}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}+\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2 \tanh (c+d x) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3}{b \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 ) a^3}{b \left (a^2+b^2\right )^{3/2} d}-\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b \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 ) a^3}{b \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 ) a^3}{b \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 ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{(e+f x)^2 \text{sech}(c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac{(e+f x)^2 a^2}{b^3 d}-\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^2}{b^3 d^2}-\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^2}{b^3 d^3}+\frac{(e+f x)^2 \tanh (c+d x) a^2}{b^3 d}-\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a}{b^2 d^2}+\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) a}{b^2 d^3}-\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) a}{b^2 d^3}+\frac{(e+f x)^2 \text{sech}(c+d x) a}{b^2 d}+\frac{(e+f x)^3}{3 b f}-\frac{(e+f x)^2}{b d}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}-\frac{(e+f x)^2 \tanh (c+d x)}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.82137, antiderivative size = 904, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 19, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.559, Rules used = {5581, 3720, 3718, 2190, 2279, 2391, 32, 5567, 5451, 4180, 5583, 4184, 5573, 3322, 2264, 2531, 2282, 6589, 6742} \[ -\frac{(e+f x)^2 a^4}{b^3 \left (a^2+b^2\right ) d}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}+\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2 \tanh (c+d x) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3}{b \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 ) a^3}{b \left (a^2+b^2\right )^{3/2} d}-\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b \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 ) a^3}{b \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 ) a^3}{b \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 ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{(e+f x)^2 \text{sech}(c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac{(e+f x)^2 a^2}{b^3 d}-\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^2}{b^3 d^2}-\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^2}{b^3 d^3}+\frac{(e+f x)^2 \tanh (c+d x) a^2}{b^3 d}-\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a}{b^2 d^2}+\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) a}{b^2 d^3}-\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) a}{b^2 d^3}+\frac{(e+f x)^2 \text{sech}(c+d x) a}{b^2 d}+\frac{(e+f x)^3}{3 b f}-\frac{(e+f x)^2}{b d}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}-\frac{(e+f x)^2 \tanh (c+d x)}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5581
Rule 3720
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 32
Rule 5567
Rule 5451
Rule 4180
Rule 5583
Rule 4184
Rule 5573
Rule 3322
Rule 2264
Rule 2531
Rule 2282
Rule 6589
Rule 6742
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \tanh ^2(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a \int (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x) \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac{\int (e+f x)^2 \, dx}{b}+\frac{(2 f) \int (e+f x) \tanh (c+d x) \, dx}{b d}\\ &=-\frac{(e+f x)^2}{b d}+\frac{(e+f x)^3}{3 b f}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}+\frac{a^2 \int (e+f x)^2 \text{sech}^2(c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{(e+f x)^2 \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac{(2 a f) \int (e+f x) \text{sech}(c+d x) \, dx}{b^2 d}+\frac{(4 f) \int \frac{e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b d}\\ &=-\frac{(e+f x)^2}{b d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a^3 \int (e+f x)^2 \text{sech}^2(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)^2}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac{\left (2 a^2 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{b^3 d}+\frac{\left (2 i a f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 d^2}-\frac{\left (2 i a f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 d^2}-\frac{\left (2 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b d^2}\\ &=\frac{a^2 (e+f x)^2}{b^3 d}-\frac{(e+f x)^2}{b d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a^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}{b^3 \left (a^2+b^2\right )}-\frac{\left (2 a^3\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}{b \left (a^2+b^2\right )}-\frac{\left (4 a^2 f\right ) \int \frac{e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b^3 d}+\frac{\left (2 i a f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^3}-\frac{\left (2 i a f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^3}-\frac{f^2 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{b d^3}\\ &=\frac{a^2 (e+f x)^2}{b^3 d}-\frac{(e+f x)^2}{b d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}-\frac{2 a^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{2 i a f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 d^3}-\frac{2 i a f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 d^3}+\frac{f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b d^3}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{\left (2 a^3\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}{\left (a^2+b^2\right )^{3/2}}+\frac{\left (2 a^3\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}{\left (a^2+b^2\right )^{3/2}}-\frac{a^4 \int (e+f x)^2 \text{sech}^2(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac{a^3 \int (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )}+\frac{\left (2 a^2 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^3 d^2}\\ &=\frac{a^2 (e+f x)^2}{b^3 d}-\frac{(e+f x)^2}{b d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac{2 a^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{2 i a f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 d^3}-\frac{2 i a f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 d^3}+\frac{f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b d^3}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a^4 (e+f x)^2 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (2 a^3 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}{b \left (a^2+b^2\right )^{3/2} d}-\frac{\left (2 a^3 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}{b \left (a^2+b^2\right )^{3/2} d}+\frac{\left (2 a^4 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (2 a^3 f\right ) \int (e+f x) \text{sech}(c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac{\left (a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{b^3 d^3}\\ &=\frac{a^2 (e+f x)^2}{b^3 d}-\frac{(e+f x)^2}{b d}-\frac{a^4 (e+f x)^2}{b^3 \left (a^2+b^2\right ) d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}+\frac{4 a^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac{2 a^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{2 i a f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 d^3}-\frac{2 i a f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 d^3}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{a^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac{f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b d^3}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a^4 (e+f x)^2 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (4 a^4 f\right ) \int \frac{e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (2 a^3 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}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{\left (2 a^3 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}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{\left (2 i a^3 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}+\frac{\left (2 i a^3 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}\\ &=\frac{a^2 (e+f x)^2}{b^3 d}-\frac{(e+f x)^2}{b d}-\frac{a^4 (e+f x)^2}{b^3 \left (a^2+b^2\right ) d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}+\frac{4 a^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac{2 a^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{2 a^4 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{2 i a f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 d^3}-\frac{2 i a f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 d^3}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{a^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac{f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b d^3}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a^4 (e+f x)^2 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac{\left (2 a^3 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 )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{\left (2 a^3 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 )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{\left (2 i a^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac{\left (2 i a^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac{\left (2 a^4 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^2}\\ &=\frac{a^2 (e+f x)^2}{b^3 d}-\frac{(e+f x)^2}{b d}-\frac{a^4 (e+f x)^2}{b^3 \left (a^2+b^2\right ) d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}+\frac{4 a^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac{2 a^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{2 a^4 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{2 i a f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 d^3}-\frac{2 i a^3 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac{2 i a f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 d^3}+\frac{2 i a^3 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{a^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac{f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b d^3}+\frac{2 a^3 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 a^3 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a^4 (e+f x)^2 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}-\frac{\left (a^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^3}\\ &=\frac{a^2 (e+f x)^2}{b^3 d}-\frac{(e+f x)^2}{b d}-\frac{a^4 (e+f x)^2}{b^3 \left (a^2+b^2\right ) d}+\frac{(e+f x)^3}{3 b f}-\frac{4 a f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 d^2}+\frac{4 a^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}+\frac{a^3 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d}-\frac{2 a^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 d^2}+\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac{2 a^4 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac{2 i a f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 d^3}-\frac{2 i a^3 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac{2 i a f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 d^3}+\frac{2 i a^3 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac{2 a^3 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac{a^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b^3 d^3}+\frac{f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b d^3}+\frac{a^4 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac{2 a^3 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 a^3 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac{a (e+f x)^2 \text{sech}(c+d x)}{b^2 d}-\frac{a^3 (e+f x)^2 \text{sech}(c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac{a^2 (e+f x)^2 \tanh (c+d x)}{b^3 d}-\frac{(e+f x)^2 \tanh (c+d x)}{b d}-\frac{a^4 (e+f x)^2 \tanh (c+d x)}{b^3 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 8.54056, size = 937, normalized size = 1.04 \[ \frac{\left (2 e^2 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right ) d^2-f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^2-2 e f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^2+f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^2+2 e f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^2-2 f (e+f x) \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d+2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d+2 f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac{2 f^2 \left (-\frac{2 \tan ^{-1}\left (\frac{\sinh (c)+\cosh (c) \tanh \left (\frac{d x}{2}\right )}{\sqrt{\cosh ^2(c)-\sinh ^2(c)}}\right ) \tanh ^{-1}(\coth (c))}{\sqrt{\cosh ^2(c)-\sinh ^2(c)}}-\frac{i \text{csch}(c) \left (i \left (d x+\tanh ^{-1}(\coth (c))\right ) \left (\log \left (1-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\log \left (1+e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )+i \left (\text{PolyLog}\left (2,-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\text{PolyLog}\left (2,e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )\right )}{\sqrt{1-\coth ^2(c)}}\right ) a}{\left (a^2+b^2\right ) d^3}-\frac{4 e f \tan ^{-1}\left (\frac{\sinh (c)+\cosh (c) \tanh \left (\frac{d x}{2}\right )}{\sqrt{\cosh ^2(c)-\sinh ^2(c)}}\right ) a}{\left (a^2+b^2\right ) d^2 \sqrt{\cosh ^2(c)-\sinh ^2(c)}}+\frac{x \left (3 e^2+3 f x e+f^2 x^2\right )}{3 b}+\frac{\text{sech}(c) \text{sech}(c+d x) \left (a \cosh (c) e^2-b \sinh (d x) e^2+2 a f x \cosh (c) e-2 b f x \sinh (d x) e+a f^2 x^2 \cosh (c)-b f^2 x^2 \sinh (d x)\right )}{\left (a^2+b^2\right ) d}-\frac{b f^2 \text{csch}(c) \left (\frac{i \coth (c) \left (-d x \left (2 i \tanh ^{-1}(\coth (c))-\pi \right )-\pi \log \left (1+e^{2 d x}\right )-2 \left (i d x+i \tanh ^{-1}(\coth (c))\right ) \log \left (1-e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )+\pi \log (\cosh (d x))+2 i \tanh ^{-1}(\coth (c)) \log \left (i \sinh \left (d x+\tanh ^{-1}(\coth (c))\right )\right )+i \text{PolyLog}\left (2,e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )\right )}{\sqrt{1-\coth ^2(c)}}-d^2 e^{-\tanh ^{-1}(\coth (c))} x^2\right ) \text{sech}(c)}{\left (a^2+b^2\right ) d^3 \sqrt{\text{csch}^2(c) \left (\sinh ^2(c)-\cosh ^2(c)\right )}}+\frac{2 b e f \text{sech}(c) (\cosh (c) \log (\cosh (c) \cosh (d x)+\sinh (c) \sinh (d x))-d x \sinh (c))}{\left (a^2+b^2\right ) d^2 \left (\cosh ^2(c)-\sinh ^2(c)\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.651, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}\sinh \left ( dx+c \right ) \left ( \tanh \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 [C] time = 4.13807, size = 9671, normalized size = 10.7 \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 )^{2} \sinh{\left (c + d x \right )} \tanh ^{2}{\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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