Integrand size = 36, antiderivative size = 914 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d^2}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d} \]
[Out]
Time = 1.47 (sec) , antiderivative size = 914, normalized size of antiderivative = 1.00, number of steps used = 51, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.694, Rules used = {5708, 5569, 4269, 3797, 2221, 2317, 2438, 2702, 327, 213, 5570, 6873, 12, 6874, 6408, 4267, 2611, 2320, 6724, 4265, 5692, 3403, 2296, 3799, 5559} \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {(e+f x)^2 \text {sech}(c+d x) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 b^2}{a \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {(e+f x)^2 \tanh (c+d x) b^2}{a \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right ) b}{a^2 d^2}+\frac {2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right ) b}{a^2 d}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) b}{a^2 d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b}{a^2 d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b}{a^2 d^3}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) b}{a^2 d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) b}{a^2 d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right ) b}{a^2 d^3}-\frac {(e+f x)^2 \text {sech}(c+d x) b}{a^2 d}-\frac {2 (e+f x)^2}{a d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3} \]
[In]
[Out]
Rule 12
Rule 213
Rule 327
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 2702
Rule 3403
Rule 3797
Rule 3799
Rule 4265
Rule 4267
Rule 4269
Rule 5559
Rule 5569
Rule 5570
Rule 5692
Rule 5708
Rule 6408
Rule 6724
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = \frac {4 \int (e+f x)^2 \text {csch}^2(2 c+2 d x) \, dx}{a}-\frac {b \int (e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2} \\ & = \frac {b (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {(2 b f) \int (e+f x) \left (-\frac {\text {arctanh}(\cosh (c+d x))}{d}+\frac {\text {sech}(c+d x)}{d}\right ) \, dx}{a^2}+\frac {(4 f) \int (e+f x) \coth (2 c+2 d x) \, dx}{a d} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \int \left (a (e+f x)^2 \text {sech}^2(c+d x)-b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {\left (2 b^4\right ) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {(2 b f) \int \frac {(e+f x) (-\text {arctanh}(\cosh (c+d x))+\text {sech}(c+d x))}{d} \, dx}{a^2}-\frac {(8 f) \int \frac {e^{2 (2 c+2 d x)} (e+f x)}{1-e^{2 (2 c+2 d x)}} \, dx}{a d} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {\left (2 b^5\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^{3/2}}-\frac {\left (2 b^5\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^{3/2}}+\frac {b^2 \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {(2 b f) \int (e+f x) (-\text {arctanh}(\cosh (c+d x))+\text {sech}(c+d x)) \, dx}{a^2 d}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{2 (2 c+2 d x)}\right ) \, dx}{a d^2} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}+\frac {(2 b f) \int (-((e+f x) \text {arctanh}(\cosh (c+d x)))+(e+f x) \text {sech}(c+d x)) \, dx}{a^2 d}-\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right ) d}-\frac {\left (2 b^3 f\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {f^2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (2 c+2 d x)}\right )}{2 a d^3} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac {(2 b f) \int (e+f x) \text {arctanh}(\cosh (c+d x)) \, dx}{a^2 d}+\frac {(2 b f) \int (e+f x) \text {sech}(c+d x) \, dx}{a^2 d}-\frac {\left (4 b^2 f\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right ) d}-\frac {\left (2 b^4 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 b^4 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 i b^3 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (2 i b^3 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d^2}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b \int d (e+f x)^2 \text {csch}(c+d x) \, dx}{a^2 d}-\frac {\left (2 b^4 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 )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 b^4 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 )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 i b^3 f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (2 i b^3 f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (2 i b f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac {\left (2 i b f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac {\left (2 b^2 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d^2} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d^2}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a^2}-\frac {\left (2 i b f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}+\frac {\left (2 i b f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d^2}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}-\frac {2 i b f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}+\frac {(2 b f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d}-\frac {(2 b f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d^2}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac {\left (2 b f^2\right ) \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac {\left (2 b f^2\right ) \int \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a^2 d^2} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d^2}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}+\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3} \\ & = -\frac {2 (e+f x)^2}{a d}+\frac {b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d^2}-\frac {4 b^3 f (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b^3 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b^4 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b^4 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x)^2 \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2029\) vs. \(2(914)=1828\).
Time = 9.33 (sec) , antiderivative size = 2029, normalized size of antiderivative = 2.22 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{2} \operatorname {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 10432 vs. \(2 (845) = 1690\).
Time = 0.51 (sec) , antiderivative size = 10432, normalized size of antiderivative = 11.41 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{2} \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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