Integrand size = 34, antiderivative size = 1176 \[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \arctan (\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \arctan (\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (e+f x)^2 \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 b^2 (e+f x)^2 \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 b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 b f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 b f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a b^2 f (e+f x) \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 {2 a b^2 f (e+f x) \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 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 b f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a^2 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 b f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a^2 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {2 a b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {a b^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d} \]
[Out]
Time = 1.25 (sec) , antiderivative size = 1176, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {5702, 4271, 3855, 4265, 2611, 2320, 6724, 5692, 5680, 2221, 6874, 3799, 5559, 4269, 3556} \[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^2 \arctan \left (e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d}-\frac {2 b (e+f x)^2 \arctan \left (e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d}+\frac {f^2 \arctan (\sinh (c+d x)) a^2}{b \left (a^2+b^2\right ) d^3}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac {i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}-\frac {2 i b f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}+\frac {i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}-\frac {f (e+f x) \text {sech}(c+d x) a^2}{b \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) a^2}{2 b \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {sech}^2(c+d x) a}{2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a}{\left (a^2+b^2\right )^2 d}-\frac {b^2 (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a}{\left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{\left (a^2+b^2\right )^2 d}-\frac {f^2 \log (\cosh (c+d x)) a}{\left (a^2+b^2\right ) d^3}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) a}{2 \left (a^2+b^2\right )^2 d^3}+\frac {f (e+f x) \tanh (c+d x) a}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {f^2 \arctan (\sinh (c+d x))}{b d^3}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}+\frac {i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d} \]
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3556
Rule 3799
Rule 3855
Rule 4265
Rule 4269
Rule 4271
Rule 5559
Rule 5680
Rule 5692
Rule 5702
Rule 6724
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = \frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac {\int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 b}-\frac {a \int (e+f x)^2 \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{b \left (a^2+b^2\right )}-\frac {(a b) \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}-\frac {f^2 \int \text {sech}(c+d x) \, dx}{b d^2} \\ & = \frac {(e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {f^2 \arctan (\sinh (c+d x))}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {(a b) \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a b^3\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2}-\frac {a \int \left (a (e+f x)^2 \text {sech}^3(c+d x)-b (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{b \left (a^2+b^2\right )}-\frac {(i f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b d}+\frac {(i f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b d} \\ & = \frac {a b^2 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac {(e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {f^2 \arctan (\sinh (c+d x))}{b d^3}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {(a b) \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a b^3\right ) \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a b^3\right ) \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac {a \int (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a^2+b^2}-\frac {a^2 \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{b \left (a^2+b^2\right )}+\frac {\left (i f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{b d^2}-\frac {\left (i f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{b d^2} \\ & = \frac {a b^2 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac {(e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {f^2 \arctan (\sinh (c+d x))}{b d^3}-\frac {a b^2 (e+f x)^2 \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 b^2 (e+f x)^2 \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 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac {\left (a^2 b\right ) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a b^2\right ) \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {a^2 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 b \left (a^2+b^2\right )}+\frac {\left (2 a b^2 f\right ) \int (e+f x) \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 (2 a b^2 f\right ) \int (e+f x) \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 {(a f) \int (e+f x) \text {sech}^2(c+d x) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (i f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}-\frac {\left (i f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}+\frac {\left (a^2 f^2\right ) \int \text {sech}(c+d x) \, dx}{b \left (a^2+b^2\right ) d^2} \\ & = \frac {(e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \arctan (\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \arctan (\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (e+f x)^2 \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 b^2 (e+f x)^2 \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 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {2 a b^2 f (e+f x) \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 {2 a b^2 f (e+f x) \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 {i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {\left (2 a b^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (2 i a^2 b f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 i a^2 b f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {\left (i a^2 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d}-\frac {\left (i a^2 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d}+\frac {\left (2 a b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (2 a b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (a f^2\right ) \int \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d^2} \\ & = \frac {(e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \arctan (\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \arctan (\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (e+f x)^2 \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 b^2 (e+f x)^2 \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 b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 b f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 b f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a b^2 f (e+f x) \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 {2 a b^2 f (e+f x) \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 {i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac {\left (2 a b^2 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {\left (2 a b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\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^3}+\frac {\left (2 a b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\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^3}-\frac {\left (2 i a^2 b f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (2 i a^2 b f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (i a^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}+\frac {\left (i a^2 f^2\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2} \\ & = \text {Too large to display} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3390\) vs. \(2(1176)=2352\).
Time = 12.13 (sec) , antiderivative size = 3390, normalized size of antiderivative = 2.88 \[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{2} \operatorname {sech}\left (d x +c \right )^{2} \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 11164 vs. \(2 (1078) = 2156\).
Time = 0.45 (sec) , antiderivative size = 11164, normalized size of antiderivative = 9.49 \[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \tanh {\left (c + d x \right )} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {sech}\left (d x + c\right )^{2} \tanh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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