Integrand size = 26, antiderivative size = 894 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^4 f \text {sech}(c+d x)}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a^3 (e+f x) \text {sech}^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^3 f \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^4 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 \left (a^2+b^2\right ) d} \]
[Out]
Time = 0.95 (sec) , antiderivative size = 894, normalized size of antiderivative = 1.00, number of steps used = 55, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {5686, 5563, 4265, 2317, 2438, 4270, 5702, 5559, 3852, 8, 5692, 5680, 2221, 6874, 3799} \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x) \arctan \left (e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d}-\frac {2 (e+f x) \arctan \left (e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^4}{b \left (a^2+b^2\right )^2 d^2}-\frac {f \text {sech}(c+d x) a^4}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x) a^4}{2 b^3 \left (a^2+b^2\right ) d}-\frac {(e+f x) \text {sech}^2(c+d x) a^3}{2 b^2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) 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 ) a^3}{\left (a^2+b^2\right )^2 d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^3}{\left (a^2+b^2\right )^2 d}-\frac {f \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-\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 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^3}{2 \left (a^2+b^2\right )^2 d^2}+\frac {f \tanh (c+d x) a^3}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \arctan \left (e^{c+d x}\right ) a^2}{b^3 d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^2}{2 b^3 d^2}+\frac {f \text {sech}(c+d x) a^2}{2 b^3 d^2}+\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x) a^2}{2 b^3 d}+\frac {(e+f x) \text {sech}^2(c+d x) a}{2 b^2 d}-\frac {f \tanh (c+d x) a}{2 b^2 d^2}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d} \]
[In]
[Out]
Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 3852
Rule 4265
Rule 4270
Rule 5559
Rule 5563
Rule 5680
Rule 5686
Rule 5692
Rule 5702
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \text {sech}(c+d x) \tanh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {a \int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {\int (e+f x) \text {sech}(c+d x) \, dx}{b}-\frac {\int (e+f x) \text {sech}^3(c+d x) \, dx}{b} \\ & = \frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {f \text {sech}(c+d x)}{2 b d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac {a^2 \int (e+f x) \text {sech}^3(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac {\int (e+f x) \text {sech}(c+d x) \, dx}{2 b}-\frac {(a f) \int \text {sech}^2(c+d x) \, dx}{2 b^2 d}-\frac {(i f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b d}+\frac {(i f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b d} \\ & = \frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac {a^2 \int (e+f x) \text {sech}(c+d x) \, dx}{2 b^3}-\frac {a^3 \int (e+f x) \text {sech}^3(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) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac {(i a f) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 b^2 d^2}-\frac {(i f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^2}+\frac {(i f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^2}+\frac {(i f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 b d}-\frac {(i f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 b d} \\ & = \frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^3 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{b \left (a^2+b^2\right )^2}-\frac {\left (a^3 b\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2}-\frac {a^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}{b^3 \left (a^2+b^2\right )}+\frac {(i f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b d^2}-\frac {(i f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b d^2}-\frac {\left (i a^2 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 b^3 d}+\frac {\left (i a^2 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 b^3 d} \\ & = \frac {a^3 (e+f x)^2}{2 \left (a^2+b^2\right )^2 f}+\frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^3 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{b \left (a^2+b^2\right )^2}-\frac {\left (a^3 b\right ) \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a^3 b\right ) \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}-\frac {a^4 \int (e+f x) \text {sech}^3(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac {a^3 \int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {\left (i a^2 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^3 d^2}+\frac {\left (i a^2 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^3 d^2} \\ & = \frac {a^3 (e+f x)^2}{2 \left (a^2+b^2\right )^2 f}+\frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^4 f \text {sech}(c+d x)}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a^3 (e+f x) \text {sech}^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^4 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 \left (a^2+b^2\right ) d}+\frac {a^3 \int (e+f x) \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {a^4 \int (e+f x) \text {sech}(c+d x) \, dx}{b \left (a^2+b^2\right )^2}-\frac {a^4 \int (e+f x) \text {sech}(c+d x) \, dx}{2 b^3 \left (a^2+b^2\right )}+\frac {\left (a^3 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {\left (a^3 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {\left (a^3 f\right ) \int \text {sech}^2(c+d x) \, dx}{2 b^2 \left (a^2+b^2\right ) d} \\ & = \frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^4 f \text {sech}(c+d x)}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a^3 (e+f x) \text {sech}^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^4 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 \left (a^2+b^2\right ) d}+\frac {\left (2 a^3\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^3 f\right ) \text {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 )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (a^3 f\right ) \text {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 )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (i a^3 f\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (i a^4 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right )^2 d}-\frac {\left (i a^4 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right )^2 d}+\frac {\left (i a^4 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 b^3 \left (a^2+b^2\right ) d}-\frac {\left (i a^4 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 b^3 \left (a^2+b^2\right ) d} \\ & = \frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^4 f \text {sech}(c+d x)}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a^3 (e+f x) \text {sech}^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^3 f \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^4 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 \left (a^2+b^2\right ) d}+\frac {\left (i a^4 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}-\frac {\left (i a^4 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}+\frac {\left (i a^4 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {\left (i a^4 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {\left (a^3 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d} \\ & = \frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^4 f \text {sech}(c+d x)}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a^3 (e+f x) \text {sech}^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^3 f \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^4 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 \left (a^2+b^2\right ) d}-\frac {\left (a^3 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2} \\ & = \frac {a^2 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 d}+\frac {(e+f x) \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {a^4 (e+f x) \arctan \left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {i a^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 d^2}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}+\frac {i a^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {i a^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right )^2 d^2}-\frac {i a^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 b^3 \left (a^2+b^2\right ) d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a^2 f \text {sech}(c+d x)}{2 b^3 d^2}-\frac {f \text {sech}(c+d x)}{2 b d^2}-\frac {a^4 f \text {sech}(c+d x)}{2 b^3 \left (a^2+b^2\right ) d^2}+\frac {a (e+f x) \text {sech}^2(c+d x)}{2 b^2 d}-\frac {a^3 (e+f x) \text {sech}^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a f \tanh (c+d x)}{2 b^2 d^2}+\frac {a^3 f \tanh (c+d x)}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 d}-\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^4 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 b^3 \left (a^2+b^2\right ) d} \\ \end{align*}
Time = 9.10 (sec) , antiderivative size = 834, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {-2 a^3 d e (c+d x)+2 a^3 c f (c+d x)-a^3 f (c+d x)^2+6 a^2 b d e \arctan \left (e^{c+d x}\right )+2 b^3 d e \arctan \left (e^{c+d x}\right )-6 a^2 b c f \arctan \left (e^{c+d x}\right )-2 b^3 c f \arctan \left (e^{c+d x}\right )+3 i a^2 b f (c+d x) \log \left (1-i e^{c+d x}\right )+i b^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i a^2 b f (c+d x) \log \left (1+i e^{c+d x}\right )-i b^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a^3 d e \log \left (1+e^{2 (c+d x)}\right )-2 a^3 c f \log \left (1+e^{2 (c+d x)}\right )+2 a^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b \left (3 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+i b \left (3 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (-b f-a f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (a d e-a c f+a f (c+d x)-b d e \sinh (c+d x)+b c f \sinh (c+d x)-b f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2283 vs. \(2 (820 ) = 1640\).
Time = 2.71 (sec) , antiderivative size = 2284, normalized size of antiderivative = 2.55
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4729 vs. \(2 (793) = 1586\).
Time = 0.37 (sec) , antiderivative size = 4729, normalized size of antiderivative = 5.29 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \tanh ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \tanh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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