Integrand size = 28, antiderivative size = 721 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^4} \]
[Out]
Time = 1.14 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {5688, 3801, 3797, 2221, 2611, 2320, 6724, 32, 5704, 5558, 3377, 2717, 4267, 6744, 5684, 3403, 2296} \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}+\frac {6 f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {(e+f x)^3}{a d} \]
[In]
[Out]
Rule 32
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 2717
Rule 3377
Rule 3403
Rule 3797
Rule 3801
Rule 4267
Rule 5558
Rule 5684
Rule 5688
Rule 5704
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \coth ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\int (e+f x)^3 \, dx}{a}-\frac {b \int (e+f x)^3 \cosh (c+d x) \coth (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {(3 f) \int (e+f x)^2 \coth (c+d x) \, dx}{a d} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {\int (e+f x)^3 \, dx}{a}-\frac {b \int (e+f x)^3 \text {csch}(c+d x) \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(6 f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a d} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {\left (2 \left (a^2+b^2\right )\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^2}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}+\frac {\left (2 b \sqrt {a^2+b^2}\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^2}-\frac {\left (2 b \sqrt {a^2+b^2}\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^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac {\left (6 b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx}{a d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (3 \sqrt {a^2+b^2} 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^2 d}+\frac {\left (3 \sqrt {a^2+b^2} 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^2 d}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a^2 d^3}-\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx}{a^2 d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}-\frac {\left (6 \sqrt {a^2+b^2} 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^2 d^2}+\frac {\left (6 \sqrt {a^2+b^2} 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^2 d^2}+\frac {\left (6 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}-\frac {\left (6 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 \sqrt {a^2+b^2} 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^2 d^3}-\frac {\left (6 \sqrt {a^2+b^2} 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^2 d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 \sqrt {a^2+b^2} 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^2 d^4}-\frac {\left (6 \sqrt {a^2+b^2} 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^2 d^4} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1490\) vs. \(2(721)=1442\).
Time = 7.17 (sec) , antiderivative size = 1490, normalized size of antiderivative = 2.07 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {-d^3 e^2 \left (-1+e^{2 c}\right ) (b d e-3 a f) x+d^3 e^2 \left (-1+e^{2 c}\right ) (b d e+3 a f) x+2 a d^3 (e+f x)^3+3 d^2 e \left (-1+e^{2 c}\right ) f (b d e-2 a f) x \log \left (1-e^{-c-d x}\right )+3 d^2 \left (-1+e^{2 c}\right ) f^2 (b d e-a f) x^2 \log \left (1-e^{-c-d x}\right )+b d^3 \left (-1+e^{2 c}\right ) f^3 x^3 \log \left (1-e^{-c-d x}\right )-3 d^2 e \left (-1+e^{2 c}\right ) f (b d e+2 a f) x \log \left (1+e^{-c-d x}\right )-3 d^2 \left (-1+e^{2 c}\right ) f^2 (b d e+a f) x^2 \log \left (1+e^{-c-d x}\right )-b d^3 \left (-1+e^{2 c}\right ) f^3 x^3 \log \left (1+e^{-c-d x}\right )+d^2 e^2 \left (-1+e^{2 c}\right ) (b d e-3 a f) \log \left (1-e^{c+d x}\right )-d^2 e^2 \left (-1+e^{2 c}\right ) (b d e+3 a f) \log \left (1+e^{c+d x}\right )+3 d e \left (-1+e^{2 c}\right ) f (b d e+2 a f) \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+6 d \left (-1+e^{2 c}\right ) f^2 (b d e+a f) x \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+3 b d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-3 d e \left (-1+e^{2 c}\right ) f (b d e-2 a f) \operatorname {PolyLog}\left (2,e^{-c-d x}\right )-6 d \left (-1+e^{2 c}\right ) f^2 (b d e-a f) x \operatorname {PolyLog}\left (2,e^{-c-d x}\right )-3 b d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+6 \left (-1+e^{2 c}\right ) f^2 (b d e+a f) \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+6 b d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+6 \left (-1+e^{2 c}\right ) f^2 (-b d e+a f) \operatorname {PolyLog}\left (3,e^{-c-d x}\right )-6 b d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (3,e^{-c-d x}\right )+6 b \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (4,-e^{-c-d x}\right )-6 b \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (4,e^{-c-d x}\right )}{a^2 d^4 \left (-1+e^{2 c}\right )}+\frac {\sqrt {a^2+b^2} \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^2 d^4}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-e^3 \sinh \left (\frac {d x}{2}\right )-3 e^2 f x \sinh \left (\frac {d x}{2}\right )-3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )-f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^3 \sinh \left (\frac {d x}{2}\right )+3 e^2 f x \sinh \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )+f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \]
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\[\int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 4612 vs. \(2 (667) = 1334\).
Time = 0.37 (sec) , antiderivative size = 4612, normalized size of antiderivative = 6.40 \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \coth \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^3 \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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