Integrand size = 34, antiderivative size = 1164 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]
[Out]
Time = 1.56 (sec) , antiderivative size = 1164, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {5708, 2702, 327, 213, 5570, 6873, 12, 6874, 6408, 4267, 2611, 6744, 2320, 6724, 4265, 5692, 3403, 2296, 2221, 4269, 3799, 5559} \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}-\frac {(e+f x)^3 \text {sech}(c+d x) b^2}{a \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 b}{\left (a^2+b^2\right ) d}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) b}{2 \left (a^2+b^2\right ) d^4}-\frac {(e+f x)^3 \tanh (c+d x) b}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d} \]
[In]
[Out]
Rule 12
Rule 213
Rule 327
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 2702
Rule 3403
Rule 3799
Rule 4265
Rule 4267
Rule 4269
Rule 5559
Rule 5570
Rule 5692
Rule 5708
Rule 6408
Rule 6724
Rule 6744
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x)^3 \text {arctanh}(\cosh (c+d x))}{a d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b \int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac {(3 f) \int (e+f x)^2 \left (-\frac {\text {arctanh}(\cosh (c+d x))}{d}+\frac {\text {sech}(c+d x)}{d}\right ) \, dx}{a} \\ & = -\frac {(e+f x)^3 \text {arctanh}(\cosh (c+d x))}{a d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b \int \left (a (e+f x)^3 \text {sech}^2(c+d x)-b (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )}-\frac {(3 f) \int \frac {(e+f x)^2 (-\text {arctanh}(\cosh (c+d x))+\text {sech}(c+d x))}{d} \, dx}{a} \\ & = -\frac {(e+f x)^3 \text {arctanh}(\cosh (c+d x))}{a d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {\left (2 b^4\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 b^4\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^{3/2}}-\frac {b \int (e+f x)^3 \text {sech}^2(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {(3 f) \int (e+f x)^2 (-\text {arctanh}(\cosh (c+d x))+\text {sech}(c+d x)) \, dx}{a d} \\ & = -\frac {(e+f x)^3 \text {arctanh}(\cosh (c+d x))}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(3 f) \int \left (-(e+f x)^2 \text {arctanh}(\cosh (c+d x))+(e+f x)^2 \text {sech}(c+d x)\right ) \, dx}{a d}+\frac {\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d}-\frac {\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(3 b f) \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a \left (a^2+b^2\right ) d} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^3 \text {arctanh}(\cosh (c+d x))}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {(3 f) \int (e+f x)^2 \text {arctanh}(\cosh (c+d x)) \, dx}{a d}-\frac {(3 f) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a d}+\frac {(6 b f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (6 b^3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (6 b^3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (6 i b^2 f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}+\frac {\left (6 i b^2 f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\int d (-e-f x)^3 \text {csch}(c+d x) \, dx}{a d}+\frac {\left (6 i f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (6 i f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac {\left (6 b^3 f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (6 b^3 f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (6 i b^2 f^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}-\frac {\left (6 i b^2 f^3\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^3} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\int (-e-f x)^3 \text {csch}(c+d x) \, dx}{a}-\frac {\left (6 b^3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {\left (6 b^3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {\left (6 i b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {\left (6 i b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {\left (6 i f^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (6 i f^3\right ) \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (3 b f^3\right ) \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^3} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(3 f) \int (-e-f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {(3 f) \int (-e-f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (3 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (6 f^2\right ) \int (-e-f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (-e-f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a d^2} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx}{a d^3} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4} \\ & = -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \\ \end{align*}
Time = 9.07 (sec) , antiderivative size = 1441, normalized size of antiderivative = 1.24 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=4 \left (-\frac {f \text {csch}(c+d x) \left (12 b d^3 e^2 e^{2 c} x-12 b d^3 e^2 \left (1+e^{2 c}\right ) x-12 b d^3 e f x^2-4 b d^3 f^2 x^3+12 a d^2 e^2 \left (1+e^{2 c}\right ) \arctan \left (e^{c+d x}\right )+6 b d^2 e^2 \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+12 i a d e \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+\operatorname {PolyLog}\left (2,i e^{c+d x}\right )\right )+6 b d e \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )+6 i a \left (1+e^{2 c}\right ) f^2 \left (d^2 x^2 \log \left (1-i e^{c+d x}\right )-d^2 x^2 \log \left (1+i e^{c+d x}\right )-2 d x \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+2 d x \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )-2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )\right )+b \left (1+e^{2 c}\right ) f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1+e^{2 (c+d x)}\right )\right )-6 d x \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )\right ) (a+b \sinh (c+d x))}{8 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right ) (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \left ((e+f x)^3 \log \left (1-e^{c+d x}\right )-(e+f x)^3 \log \left (1+e^{c+d x}\right )-\frac {3 f \left (d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )+2 f^2 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )\right )}{d^3}+\frac {3 f \left (d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )+2 f^2 \operatorname {PolyLog}\left (4,e^{c+d x}\right )\right )}{d^3}\right ) (a+b \sinh (c+d x))}{4 a d (b+a \text {csch}(c+d x))}-\frac {b^3 \text {csch}(c+d x) \left (-2 d^3 e^3 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-6 d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+6 d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 f^3 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) (a+b \sinh (c+d x))}{4 a \left (a^2+b^2\right )^{3/2} d^4 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}(c) \text {sech}(c+d x) \left (a e^3 \cosh (c)+3 a e^2 f x \cosh (c)+3 a e f^2 x^2 \cosh (c)+a f^3 x^3 \cosh (c)-b e^3 \sinh (d x)-3 b e^2 f x \sinh (d x)-3 b e f^2 x^2 \sinh (d x)-b f^3 x^3 \sinh (d x)\right ) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d (b+a \text {csch}(c+d x))}\right ) \]
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\[\int \frac {\left (f x +e \right )^{3} \operatorname {csch}\left (d x +c \right ) \operatorname {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 9707 vs. \(2 (1064) = 2128\).
Time = 0.51 (sec) , antiderivative size = 9707, normalized size of antiderivative = 8.34 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right ) \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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