Integrand size = 28, antiderivative size = 1053 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^2}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^3}+\frac {3 b f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^3 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^4} \]
[Out]
Time = 1.22 (sec) , antiderivative size = 1053, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5694, 4271, 4267, 2317, 2438, 2611, 6744, 2320, 6724, 4269, 3797, 2221, 3403, 2296} \[ \int \frac {(e+f x)^3 \text {csch}^3(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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^4}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right ) b^2}{a^3 d}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) b^2}{a^3 d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right ) b^2}{a^3 d^2}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) b^2}{a^3 d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right ) b^2}{a^3 d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right ) b^2}{a^3 d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right ) b^2}{a^3 d^4}+\frac {(e+f x)^3 b}{a^2 d}+\frac {(e+f x)^3 \coth (c+d x) b}{a^2 d}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) b}{a^2 d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) b}{a^2 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right ) b}{2 a^2 d^4}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4} \]
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3403
Rule 3797
Rule 4267
Rule 4269
Rule 4271
Rule 5694
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \text {csch}^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int (e+f x)^3 \text {csch}(c+d x) \, dx}{2 a}-\frac {b \int (e+f x)^3 \text {csch}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {\left (3 f^2\right ) \int (e+f x) \text {csch}(c+d x) \, dx}{a d^2} \\ & = -\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {b^2 \int (e+f x)^3 \text {csch}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{2 a d}-\frac {(3 b f) \int (e+f x)^2 \coth (c+d x) \, dx}{a^2 d}-\frac {\left (3 f^3\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (3 f^3\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d^3} \\ & = \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\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^3}+\frac {(6 b f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a^2 d}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}-\frac {\left (3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4} \\ & = \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2}}+\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}+\frac {\left (6 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (6 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a^3 d^2}+\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx}{a d^3} \\ & = \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d}+\frac {\left (3 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 (3 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 (3 b f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx}{a^2 d^3}-\frac {\left (6 b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a^3 d^3}+\frac {\left (6 b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx}{a^3 d^3} \\ & = \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^2}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^2}+\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 a^2 d^4}-\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}+\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4} \\ & = \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^2}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^3}+\frac {3 b f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^3 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^3 d^4}-\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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^3} \\ & = \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^2}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^3}+\frac {3 b f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^3 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^4} \\ & = \frac {b (e+f x)^3}{a^2 d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d}-\frac {3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^2}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^3}+\frac {3 b f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^3 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^3 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^3 \sqrt {a^2+b^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^3 \sqrt {a^2+b^2} d^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(2801\) vs. \(2(1053)=2106\).
Time = 9.46 (sec) , antiderivative size = 2801, normalized size of antiderivative = 2.66 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{3} \operatorname {csch}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 18159 vs. \(2 (977) = 1954\).
Time = 0.58 (sec) , antiderivative size = 18159, normalized size of antiderivative = 17.25 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
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