Integrand size = 28, antiderivative size = 725 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3} \]
[Out]
Time = 0.93 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5694, 4271, 3855, 4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438, 3403, 2296} \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}+\frac {b (e+f x)^2}{a^2 d}+\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3 \sqrt {a^2+b^2}}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3 \sqrt {a^2+b^2}}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2 \sqrt {a^2+b^2}}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2 \sqrt {a^2+b^2}}-\frac {b^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d \sqrt {a^2+b^2}}+\frac {b^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d \sqrt {a^2+b^2}}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3403
Rule 3797
Rule 3855
Rule 4267
Rule 4269
Rule 4271
Rule 5694
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \text {csch}^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int (e+f x)^2 \text {csch}(c+d x) \, dx}{2 a}-\frac {b \int (e+f x)^2 \text {csch}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {f^2 \int \text {csch}(c+d x) \, dx}{a d^2} \\ & = \frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {b^2 \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {f \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {f \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {(2 b f) \int (e+f x) \coth (c+d x) \, dx}{a^2 d} \\ & = \frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {\left (2 b^3\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^3}+\frac {(4 b f) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^2 d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}-\frac {f^2 \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a d^2}+\frac {f^2 \int \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a d^2} \\ & = \frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {\left (2 b^4\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^3 \sqrt {a^2+b^2}}+\frac {\left (2 b^4\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^3 \sqrt {a^2+b^2}}-\frac {f^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {f^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 b f^2\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}+\frac {\left (2 b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (2 b^2 f^2\right ) \int \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a^3 d^2} \\ & = \frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {\left (2 b^3 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^3 \sqrt {a^2+b^2} d}-\frac {\left (2 b^3 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^3 \sqrt {a^2+b^2} d}+\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a^2 d^3}+\frac {\left (2 b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {\left (2 b^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3} \\ & = \frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {\left (2 b^3 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^3 \sqrt {a^2+b^2} d^2}-\frac {\left (2 b^3 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^3 \sqrt {a^2+b^2} d^2} \\ & = \frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {\left (2 b^3 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^3 \sqrt {a^2+b^2} d^3}-\frac {\left (2 b^3 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^3 \sqrt {a^2+b^2} d^3} \\ & = \frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1532\) vs. \(2(725)=1450\).
Time = 8.24 (sec) , antiderivative size = 1532, normalized size of antiderivative = 2.11 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {8 a b d^2 e e^{2 c} f x+4 a b d^2 e^{2 c} f^2 x^2-2 a^2 d^2 e^2 \text {arctanh}\left (e^{c+d x}\right )+4 b^2 d^2 e^2 \text {arctanh}\left (e^{c+d x}\right )+2 a^2 d^2 e^2 e^{2 c} \text {arctanh}\left (e^{c+d x}\right )-4 b^2 d^2 e^2 e^{2 c} \text {arctanh}\left (e^{c+d x}\right )+4 a^2 f^2 \text {arctanh}\left (e^{c+d x}\right )-4 a^2 e^{2 c} f^2 \text {arctanh}\left (e^{c+d x}\right )+2 a^2 d^2 e f x \log \left (1-e^{c+d x}\right )-4 b^2 d^2 e f x \log \left (1-e^{c+d x}\right )-2 a^2 d^2 e e^{2 c} f x \log \left (1-e^{c+d x}\right )+4 b^2 d^2 e e^{2 c} f x \log \left (1-e^{c+d x}\right )+a^2 d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )-2 b^2 d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )-a^2 d^2 e^{2 c} f^2 x^2 \log \left (1-e^{c+d x}\right )+2 b^2 d^2 e^{2 c} f^2 x^2 \log \left (1-e^{c+d x}\right )-2 a^2 d^2 e f x \log \left (1+e^{c+d x}\right )+4 b^2 d^2 e f x \log \left (1+e^{c+d x}\right )+2 a^2 d^2 e e^{2 c} f x \log \left (1+e^{c+d x}\right )-4 b^2 d^2 e e^{2 c} f x \log \left (1+e^{c+d x}\right )-a^2 d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )+2 b^2 d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )+a^2 d^2 e^{2 c} f^2 x^2 \log \left (1+e^{c+d x}\right )-2 b^2 d^2 e^{2 c} f^2 x^2 \log \left (1+e^{c+d x}\right )+4 a b d e f \log \left (1-e^{2 (c+d x)}\right )-4 a b d e e^{2 c} f \log \left (1-e^{2 (c+d x)}\right )+4 a b d f^2 x \log \left (1-e^{2 (c+d x)}\right )-4 a b d e^{2 c} f^2 x \log \left (1-e^{2 (c+d x)}\right )+2 \left (a^2-2 b^2\right ) d \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-2 \left (a^2-2 b^2\right ) d \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )+2 a b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )-2 a b e^{2 c} f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )+2 a^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )-4 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )-2 a^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )+4 b^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )-2 a^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )+4 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )+2 a^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )-4 b^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{2 a^3 d^3 \left (-1+e^{2 c}\right )}-\frac {b^3 \left (-2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 \sqrt {a^2+b^2} d^3}+\frac {\text {csch}(c) \text {csch}^2(c+d x) \left (2 b d e^2 \cosh (c)+4 b d e f x \cosh (c)+2 b d f^2 x^2 \cosh (c)+2 a e f \cosh (d x)+2 a f^2 x \cosh (d x)-2 a e f \cosh (2 c+d x)-2 a f^2 x \cosh (2 c+d x)-2 b d e^2 \cosh (c+2 d x)-4 b d e f x \cosh (c+2 d x)-2 b d f^2 x^2 \cosh (c+2 d x)+a d e^2 \sinh (d x)+2 a d e f x \sinh (d x)+a d f^2 x^2 \sinh (d x)-a d e^2 \sinh (2 c+d x)-2 a d e f x \sinh (2 c+d x)-a d f^2 x^2 \sinh (2 c+d x)\right )}{4 a^2 d^2} \]
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\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 10341 vs. \(2 (676) = 1352\).
Time = 0.40 (sec) , antiderivative size = 10341, normalized size of antiderivative = 14.26 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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