Optimal. Leaf size=978 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.411, antiderivative size = 978, normalized size of antiderivative = 1., number of steps used = 57, number of rules used = 27, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.794, Rules used = {5589, 2621, 288, 321, 207, 5462, 5203, 12, 4180, 2279, 2391, 3770, 2622, 2620, 14, 2548, 4182, 3473, 8, 5573, 5561, 2190, 6742, 3718, 4185, 5451, 3767} \[ \frac{(e+f x) \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac{(e+f x) \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}-\frac{(e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac{f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac{f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac{f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^5}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac{2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac{(e+f x) \text{sech}^2(c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d}-\frac{f \tanh (c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac{(e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d}-\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac{f \text{sech}(c+d x) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac{(e+f x) \text{sech}(c+d x) \tanh (c+d x) b^2}{2 a \left (a^2+b^2\right ) d}+\frac{(e+f x) \tanh ^2(c+d x) b}{2 a^2 d}-\frac{f x b}{2 a^2 d}+\frac{2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b}{a^2 d}+\frac{f x \log (\tanh (c+d x)) b}{a^2 d}-\frac{(e+f x) \log (\tanh (c+d x)) b}{a^2 d}+\frac{f \text{PolyLog}\left (2,-e^{2 c+2 d x}\right ) b}{2 a^2 d^2}-\frac{f \text{PolyLog}\left (2,e^{2 c+2 d x}\right ) b}{2 a^2 d^2}+\frac{f \tanh (c+d x) b}{2 a^2 d^2}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}-\frac{3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{3 i f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^2}-\frac{3 i f \text{PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^2}-\frac{f \text{sech}(c+d x)}{2 a d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5589
Rule 2621
Rule 288
Rule 321
Rule 207
Rule 5462
Rule 5203
Rule 12
Rule 4180
Rule 2279
Rule 2391
Rule 3770
Rule 2622
Rule 2620
Rule 14
Rule 2548
Rule 4182
Rule 3473
Rule 8
Rule 5573
Rule 5561
Rule 2190
Rule 6742
Rule 3718
Rule 4185
Rule 5451
Rule 3767
Rubi steps
\begin{align*} \int \frac{(e+f x) \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \text{csch}^2(c+d x) \text{sech}^3(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x) \text{csch}(c+d x) \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}-\frac{b \int (e+f x) \text{csch}(c+d x) \text{sech}^3(c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x) \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac{f \int \left (-\frac{3 \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{3 \text{csch}(c+d x)}{2 d}+\frac{\text{csch}(c+d x) \text{sech}^2(c+d x)}{2 d}\right ) \, dx}{a}\\ &=-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac{b^2 \int (e+f x) \text{sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )}+\frac{b^4 \int \frac{(e+f x) \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac{(b f) \int \left (\frac{\log (\tanh (c+d x))}{d}-\frac{\tanh ^2(c+d x)}{2 d}\right ) \, dx}{a^2}-\frac{f \int \text{csch}(c+d x) \text{sech}^2(c+d x) \, dx}{2 a d}+\frac{(3 f) \int \tan ^{-1}(\sinh (c+d x)) \, dx}{2 a d}+\frac{(3 f) \int \text{csch}(c+d x) \, dx}{2 a d}\\ &=\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 f \tanh ^{-1}(\cosh (c+d x))}{2 a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac{b^4 \int (e+f x) \text{sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac{b^6 \int \frac{(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac{b^2 \int \left (a (e+f x) \text{sech}^3(c+d x)-b (e+f x) \text{sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{a^2 \left (a^2+b^2\right )}-\frac{f \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{2 a d^2}-\frac{(3 f) \int d x \text{sech}(c+d x) \, dx}{2 a d}-\frac{(b f) \int \tanh ^2(c+d x) \, dx}{2 a^2 d}+\frac{(b f) \int \log (\tanh (c+d x)) \, dx}{a^2 d}\\ &=-\frac{b^5 (e+f x)^2}{2 a^2 \left (a^2+b^2\right )^2 f}+\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 f \tanh ^{-1}(\cosh (c+d x))}{2 a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{b f x \log (\tanh (c+d x))}{a^2 d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}-\frac{f \text{sech}(c+d x)}{2 a d^2}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b f \tanh (c+d x)}{2 a^2 d^2}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac{b^4 \int (a (e+f x) \text{sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac{b^6 \int \frac{e^{c+d x} (e+f x)}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac{b^6 \int \frac{e^{c+d x} (e+f x)}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac{b^2 \int (e+f x) \text{sech}^3(c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int (e+f x) \text{sech}^2(c+d x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )}-\frac{(3 f) \int x \text{sech}(c+d x) \, dx}{2 a}-\frac{f \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{2 a d^2}-\frac{(b f) \int 1 \, dx}{2 a^2 d}-\frac{(b f) \int 2 d x \text{csch}(2 c+2 d x) \, dx}{a^2 d}\\ &=-\frac{b f x}{2 a^2 d}-\frac{b^5 (e+f x)^2}{2 a^2 \left (a^2+b^2\right )^2 f}-\frac{3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b f x \log (\tanh (c+d x))}{a^2 d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}-\frac{f \text{sech}(c+d x)}{2 a d^2}+\frac{b^2 f \text{sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac{b^3 (e+f x) \text{sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b f \tanh (c+d x)}{2 a^2 d^2}+\frac{b^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac{b^4 \int (e+f x) \text{sech}(c+d x) \, dx}{a \left (a^2+b^2\right )^2}-\frac{b^5 \int (e+f x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac{b^2 \int (e+f x) \text{sech}(c+d x) \, dx}{2 a \left (a^2+b^2\right )}-\frac{(2 b f) \int x \text{csch}(2 c+2 d x) \, dx}{a^2}+\frac{(3 i f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 a d}-\frac{(3 i f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 a d}-\frac{\left (b^5 f\right ) \int \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}-\frac{\left (b^5 f\right ) \int \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}-\frac{\left (b^3 f\right ) \int \text{sech}^2(c+d x) \, dx}{2 a^2 \left (a^2+b^2\right ) d}\\ &=-\frac{b f x}{2 a^2 d}-\frac{3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac{2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b f x \log (\tanh (c+d x))}{a^2 d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}-\frac{f \text{sech}(c+d x)}{2 a d^2}+\frac{b^2 f \text{sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac{b^3 (e+f x) \text{sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b f \tanh (c+d x)}{2 a^2 d^2}+\frac{b^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}-\frac{\left (2 b^5\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 )^2}+\frac{(3 i f) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac{(3 i f) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac{\left (b^5 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a-\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac{\left (b^5 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac{\left (i b^3 f\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac{(b f) \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^2 d}-\frac{(b f) \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^2 d}-\frac{\left (i b^4 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac{\left (i b^4 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}-\frac{\left (i b^2 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 a \left (a^2+b^2\right ) d}+\frac{\left (i b^2 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 a \left (a^2+b^2\right ) d}\\ &=-\frac{b f x}{2 a^2 d}-\frac{3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac{2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac{b^5 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b f x \log (\tanh (c+d x))}{a^2 d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac{3 i f \text{Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac{3 i f \text{Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac{b^5 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac{b^5 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac{f \text{sech}(c+d x)}{2 a d^2}+\frac{b^2 f \text{sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac{b^3 (e+f x) \text{sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b f \tanh (c+d x)}{2 a^2 d^2}-\frac{b^3 f \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac{b^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac{(b f) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{(b f) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{\left (i b^4 f\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac{\left (i b^4 f\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac{\left (i b^2 f\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac{\left (i b^2 f\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac{\left (b^5 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac{b f x}{2 a^2 d}-\frac{3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac{2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac{b^5 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b f x \log (\tanh (c+d x))}{a^2 d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac{3 i f \text{Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac{i b^4 f \text{Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac{i b^2 f \text{Li}_2\left (-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac{3 i f \text{Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac{i b^4 f \text{Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac{i b^2 f \text{Li}_2\left (i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac{b^5 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac{b^5 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac{b f \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{b f \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{f \text{sech}(c+d x)}{2 a d^2}+\frac{b^2 f \text{sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac{b^3 (e+f x) \text{sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b f \tanh (c+d x)}{2 a^2 d^2}-\frac{b^3 f \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac{b^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}+\frac{\left (b^5 f\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right )^2 d^2}\\ &=-\frac{b f x}{2 a^2 d}-\frac{3 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac{3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac{3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac{2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{3 (e+f x) \text{csch}(c+d x)}{2 a d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b^5 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac{b^5 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac{b f x \log (\tanh (c+d x))}{a^2 d}-\frac{b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac{3 i f \text{Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac{i b^4 f \text{Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac{i b^2 f \text{Li}_2\left (-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac{3 i f \text{Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac{i b^4 f \text{Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac{i b^2 f \text{Li}_2\left (i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac{b^5 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac{b^5 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac{b^5 f \text{Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac{b f \text{Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{b f \text{Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{f \text{sech}(c+d x)}{2 a d^2}+\frac{b^2 f \text{sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac{b^3 (e+f x) \text{sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{2 a d}+\frac{b f \tanh (c+d x)}{2 a^2 d^2}-\frac{b^3 f \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac{b^2 (e+f x) \text{sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac{b (e+f x) \tanh ^2(c+d x)}{2 a^2 d}\\ \end{align*}
Mathematica [A] time = 10.9538, size = 1337, normalized size = 1.37 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.273, size = 3280, normalized size = 3.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-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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