Optimal. Leaf size=1122 \[ -\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^6}{a^3 \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^6}{a^3 \left (a^2+b^2\right )^2 d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^6}{a^3 \left (a^2+b^2\right )^2 d}-\frac {f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^6}{a^3 \left (a^2+b^2\right )^2 d^2}-\frac {f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^6}{a^3 \left (a^2+b^2\right )^2 d^2}+\frac {f \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^6}{2 a^3 \left (a^2+b^2\right )^2 d^2}-\frac {2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {i f \text {Li}_2\left (i e^{c+d x}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x) b^4}{2 a^3 \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x) b^4}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {i f \text {Li}_2\left (i e^{c+d x}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {f \text {sech}(c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x) b^2}{2 a^3 d}+\frac {f x b^2}{2 a^3 d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b^2}{a^3 d}-\frac {f x \log (\tanh (c+d x)) b^2}{a^3 d}+\frac {(e+f x) \log (\tanh (c+d x)) b^2}{a^3 d}-\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}+\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}-\frac {f \tanh (c+d x) b^2}{2 a^3 d^2}-\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x) b}{2 a^2 d}+\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right ) b}{a^2 d}-\frac {3 f x \tan ^{-1}(\sinh (c+d x)) b}{2 a^2 d}+\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x)) b}{2 a^2 d}+\frac {f \tanh ^{-1}(\cosh (c+d x)) b}{a^2 d^2}+\frac {3 (e+f x) \text {csch}(c+d x) b}{2 a^2 d}-\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right ) b}{2 a^2 d^2}+\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right ) b}{2 a^2 d^2}+\frac {f \text {sech}(c+d x) b}{2 a^2 d^2}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.81, antiderivative size = 1122, normalized size of antiderivative = 1.00, number of steps used = 65, number of rules used = 28, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {5589, 5461, 4185, 4182, 2279, 2391, 2621, 288, 321, 207, 5462, 5203, 12, 4180, 3770, 2622, 2620, 14, 2548, 3473, 8, 5573, 5561, 2190, 6742, 3718, 5451, 3767} \[ -\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^6}{a^3 \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^6}{a^3 \left (a^2+b^2\right )^2 d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^6}{a^3 \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^6}{a^3 \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^6}{a^3 \left (a^2+b^2\right )^2 d^2}+\frac {f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^6}{2 a^3 \left (a^2+b^2\right )^2 d^2}-\frac {2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {i f \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {i f \text {PolyLog}\left (2,i e^{c+d x}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {(e+f x) \text {sech}^2(c+d x) b^4}{2 a^3 \left (a^2+b^2\right ) d}+\frac {f \tanh (c+d x) b^4}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {i f \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {i f \text {PolyLog}\left (2,i e^{c+d x}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {f \text {sech}(c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x) b^2}{2 a^3 d}+\frac {f x b^2}{2 a^3 d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b^2}{a^3 d}-\frac {f x \log (\tanh (c+d x)) b^2}{a^3 d}+\frac {(e+f x) \log (\tanh (c+d x)) b^2}{a^3 d}-\frac {f \text {PolyLog}\left (2,-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}+\frac {f \text {PolyLog}\left (2,e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}-\frac {f \tanh (c+d x) b^2}{2 a^3 d^2}-\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x) b}{2 a^2 d}+\frac {3 f x \tan ^{-1}\left (e^{c+d x}\right ) b}{a^2 d}-\frac {3 f x \tan ^{-1}(\sinh (c+d x)) b}{2 a^2 d}+\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x)) b}{2 a^2 d}+\frac {f \tanh ^{-1}(\cosh (c+d x)) b}{a^2 d^2}+\frac {3 (e+f x) \text {csch}(c+d x) b}{2 a^2 d}-\frac {3 i f \text {PolyLog}\left (2,-i e^{c+d x}\right ) b}{2 a^2 d^2}+\frac {3 i f \text {PolyLog}\left (2,i e^{c+d x}\right ) b}{2 a^2 d^2}+\frac {f \text {sech}(c+d x) b}{2 a^2 d^2}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}+\frac {f \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}-\frac {f \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 14
Rule 207
Rule 288
Rule 321
Rule 2190
Rule 2279
Rule 2391
Rule 2548
Rule 2620
Rule 2621
Rule 2622
Rule 3473
Rule 3718
Rule 3767
Rule 3770
Rule 4180
Rule 4182
Rule 4185
Rule 5203
Rule 5451
Rule 5461
Rule 5462
Rule 5561
Rule 5573
Rule 5589
Rule 6742
Rubi steps
\begin {align*} \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {8 \int (e+f x) \text {csch}^3(2 c+2 d x) \, dx}{a}-\frac {b \int (e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {4 \int (e+f x) \text {csch}(2 c+2 d x) \, dx}{a}+\frac {b^2 \int (e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {(b 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^2}\\ &=\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}-\frac {b^3 \int (e+f x) \text {sech}^3(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) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\left (b^2 f\right ) \int \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right ) \, dx}{a^3}+\frac {(2 f) \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {(2 f) \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {(b f) \int \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{2 a^2 d}-\frac {(3 b f) \int \tan ^{-1}(\sinh (c+d x)) \, dx}{2 a^2 d}-\frac {(3 b f) \int \text {csch}(c+d x) \, dx}{2 a^2 d}\\ &=-\frac {3 b f x \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {3 b f \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}-\frac {b^5 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )^2}-\frac {b^7 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )^2}-\frac {b^3 \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^3 \left (a^2+b^2\right )}+\frac {f \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{a d^2}-\frac {f \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{a d^2}+\frac {(b f) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a^2 d^2}+\frac {(3 b f) \int d x \text {sech}(c+d x) \, dx}{2 a^2 d}+\frac {\left (b^2 f\right ) \int \tanh ^2(c+d x) \, dx}{2 a^3 d}-\frac {\left (b^2 f\right ) \int \log (\tanh (c+d x)) \, dx}{a^3 d}\\ &=\frac {b^6 (e+f x)^2}{2 a^3 \left (a^2+b^2\right )^2 f}-\frac {3 b f x \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {3 b f \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b^2 f x \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b f \text {sech}(c+d x)}{2 a^2 d^2}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 f \tanh (c+d x)}{2 a^3 d^2}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}-\frac {b^5 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )^2}-\frac {b^7 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^2}-\frac {b^7 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^2}-\frac {b^3 \int (e+f x) \text {sech}^3(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(3 b f) \int x \text {sech}(c+d x) \, dx}{2 a^2}+\frac {(b f) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a^2 d^2}+\frac {\left (b^2 f\right ) \int 1 \, dx}{2 a^3 d}+\frac {\left (b^2 f\right ) \int 2 d x \text {csch}(2 c+2 d x) \, dx}{a^3 d}\\ &=\frac {b^2 f x}{2 a^3 d}+\frac {b^6 (e+f x)^2}{2 a^3 \left (a^2+b^2\right )^2 f}+\frac {3 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {3 b f x \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^2 f x \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b f \text {sech}(c+d x)}{2 a^2 d^2}-\frac {b^3 f \text {sech}(c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 (e+f x) \text {sech}^2(c+d x)}{2 a^3 \left (a^2+b^2\right ) d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 f \tanh (c+d x)}{2 a^3 d^2}-\frac {b^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}-\frac {b^5 \int (e+f x) \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right )^2}+\frac {b^6 \int (e+f x) \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )^2}-\frac {b^3 \int (e+f x) \text {sech}(c+d x) \, dx}{2 a^2 \left (a^2+b^2\right )}+\frac {\left (2 b^2 f\right ) \int x \text {csch}(2 c+2 d x) \, dx}{a^3}-\frac {(3 i b f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 a^2 d}+\frac {(3 i b f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 a^2 d}+\frac {\left (b^6 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^2 d}+\frac {\left (b^6 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^2 d}+\frac {\left (b^4 f\right ) \int \text {sech}^2(c+d x) \, dx}{2 a^3 \left (a^2+b^2\right ) d}\\ &=\frac {b^2 f x}{2 a^3 d}+\frac {3 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^5 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {3 b f x \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^2 f x \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b f \text {sech}(c+d x)}{2 a^2 d^2}-\frac {b^3 f \text {sech}(c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 (e+f x) \text {sech}^2(c+d x)}{2 a^3 \left (a^2+b^2\right ) d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 f \tanh (c+d x)}{2 a^3 d^2}-\frac {b^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}+\frac {\left (2 b^6\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^2}-\frac {(3 i b f) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a^2 d^2}+\frac {(3 i b f) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a^2 d^2}+\frac {\left (b^6 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^3 \left (a^2+b^2\right )^2 d^2}+\frac {\left (b^6 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^3 \left (a^2+b^2\right )^2 d^2}+\frac {\left (i b^4 f\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {\left (b^2 f\right ) \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (b^2 f\right ) \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (i b^5 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}-\frac {\left (i b^5 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right )^2 d}+\frac {\left (i b^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 a^2 \left (a^2+b^2\right ) d}-\frac {\left (i b^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 a^2 \left (a^2+b^2\right ) d}\\ &=\frac {b^2 f x}{2 a^3 d}+\frac {3 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^5 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {3 b f x \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}+\frac {b^6 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^2 f x \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}-\frac {3 i b f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a^2 d^2}+\frac {3 i b f \text {Li}_2\left (i e^{c+d x}\right )}{2 a^2 d^2}-\frac {b^6 f \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 )^2 d^2}-\frac {b^6 f \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 )^2 d^2}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b f \text {sech}(c+d x)}{2 a^2 d^2}-\frac {b^3 f \text {sech}(c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 (e+f x) \text {sech}^2(c+d x)}{2 a^3 \left (a^2+b^2\right ) d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 f \tanh (c+d x)}{2 a^3 d^2}+\frac {b^4 f \tanh (c+d x)}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}-\frac {\left (b^2 f\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {\left (b^2 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {\left (i b^5 f\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {\left (i b^5 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{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}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (i b^3 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (b^6 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right )^2 d}\\ &=\frac {b^2 f x}{2 a^3 d}+\frac {3 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^5 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {3 b f x \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}+\frac {b^6 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^2 f x \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}-\frac {3 i b f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a^2 d^2}+\frac {i b^5 f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f \text {Li}_2\left (i e^{c+d x}\right )}{2 a^2 d^2}-\frac {i b^5 f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^6 f \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 )^2 d^2}-\frac {b^6 f \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 )^2 d^2}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {b f \text {sech}(c+d x)}{2 a^2 d^2}-\frac {b^3 f \text {sech}(c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 (e+f x) \text {sech}^2(c+d x)}{2 a^3 \left (a^2+b^2\right ) d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 f \tanh (c+d x)}{2 a^3 d^2}+\frac {b^4 f \tanh (c+d x)}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}-\frac {\left (b^6 f\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right )^2 d^2}\\ &=\frac {b^2 f x}{2 a^3 d}+\frac {3 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^5 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {3 b f x \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}+\frac {3 b (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a^2 d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {4 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac {3 b (e+f x) \text {csch}(c+d x)}{2 a^2 d}-\frac {f \text {csch}(2 c+2 d x)}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{a d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^6 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^2 d}+\frac {b^6 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b^2 f x \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (e+f x) \log (\tanh (c+d x))}{a^3 d}-\frac {3 i b f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a^2 d^2}+\frac {i b^5 f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f \text {Li}_2\left (i e^{c+d x}\right )}{2 a^2 d^2}-\frac {i b^5 f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^6 f \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 )^2 d^2}-\frac {b^6 f \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 )^2 d^2}+\frac {b^6 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right )^2 d^2}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {b f \text {sech}(c+d x)}{2 a^2 d^2}-\frac {b^3 f \text {sech}(c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 (e+f x) \text {sech}^2(c+d x)}{2 a^3 \left (a^2+b^2\right ) d}-\frac {b (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a^2 d}-\frac {b^2 f \tanh (c+d x)}{2 a^3 d^2}+\frac {b^4 f \tanh (c+d x)}{2 a^3 \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x) \tanh ^2(c+d x)}{2 a^3 d}\\ \end {align*}
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Mathematica [B] time = 10.10, size = 2870, normalized size = 2.56 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 3563, normalized size = 3.18 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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