Integrand size = 28, antiderivative size = 972 \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {6 b f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}-\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4}+\frac {3 b^2 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^3 d^4} \]
[Out]
Time = 1.51 (sec) , antiderivative size = 972, normalized size of antiderivative = 1.00, number of steps used = 62, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.821, Rules used = {5688, 3801, 3797, 2221, 2317, 2438, 32, 2611, 6744, 2320, 6724, 5704, 5558, 3377, 2718, 5560, 4267, 5554, 3392, 2715, 8, 5684, 5680} \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}-\frac {(e+f x)^4}{4 a f}-\frac {\coth ^2(c+d x) (e+f x)^3}{2 a d}+\frac {b \text {csch}(c+d x) (e+f x)^3}{a^2 d}-\frac {\left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^3 d}-\frac {\left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^3 d}+\frac {b^2 \log \left (1-e^{2 (c+d x)}\right ) (e+f x)^3}{a^3 d}+\frac {\log \left (1-e^{2 (c+d x)}\right ) (e+f x)^3}{a d}+\frac {(e+f x)^3}{2 a d}-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {6 b f \text {arctanh}\left (e^{c+d x}\right ) (e+f x)^2}{a^2 d^2}-\frac {3 f \coth (c+d x) (e+f x)^2}{2 a d^2}-\frac {3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^3 d^2}+\frac {3 b^2 f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) (e+f x)^2}{2 a^3 d^2}+\frac {3 f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) (e+f x)^2}{2 a d^2}+\frac {3 f^2 \log \left (1-e^{2 (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 b f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) (e+f x)}{a^2 d^3}-\frac {6 b f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right ) (e+f x)}{a^2 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)}{a^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)}{a^3 d^3}-\frac {3 b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right ) (e+f x)}{2 a^3 d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right ) (e+f x)}{2 a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {3 b^2 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^3 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4} \]
[In]
[Out]
Rule 8
Rule 32
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3797
Rule 3801
Rule 4267
Rule 5554
Rule 5558
Rule 5560
Rule 5680
Rule 5684
Rule 5688
Rule 5704
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \coth ^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {\int (e+f x)^3 \coth (c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cosh (c+d x) \coth ^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {(3 f) \int (e+f x)^2 \coth ^2(c+d x) \, dx}{2 a d} \\ & = -\frac {(e+f x)^4}{4 a f}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}-\frac {2 \int \frac {e^{2 (c+d x)} (e+f x)^3}{1-e^{2 (c+d x)}} \, dx}{a}-\frac {b \int (e+f x)^3 \cosh (c+d x) \, dx}{a^2}-\frac {b \int (e+f x)^3 \coth (c+d x) \text {csch}(c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \cosh ^2(c+d x) \coth (c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {(3 f) \int (e+f x)^2 \, dx}{2 a d}+\frac {\left (3 f^2\right ) \int (e+f x) \coth (c+d x) \, dx}{a d^2} \\ & = -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {b (e+f x)^3 \sinh (c+d x)}{a^2 d}+\frac {b \int (e+f x)^3 \cosh (c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \coth (c+d x) \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d}-\frac {(3 b f) \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a^2 d}+\frac {(3 b f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a^2 d}-\frac {\left (6 f^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a d^2} \\ & = -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {3 b f (e+f x)^2 \cosh (c+d x)}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}-\frac {\left (2 b^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1-e^{2 (c+d x)}} \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3}-\frac {(3 b f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a^2 d}-\frac {\left (3 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a^2 d^2}+\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (3 f^3\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^3} \\ & = -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {6 b f^2 (e+f x) \sinh (c+d x)}{a^2 d^3}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^3 d}+\frac {\left (3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}+\frac {\left (3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}+\frac {\left (6 b f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a^2 d^2}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right ) \, dx}{2 a d^3}-\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a^2 d^3}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a^2 d^3}+\frac {\left (6 b f^3\right ) \int \sinh (c+d x) \, dx}{a^2 d^3} \\ & = -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {6 b f^3 \cosh (c+d x)}{a^2 d^4}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {\left (3 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx}{a^3 d^2}+\frac {\left (6 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^2}+\frac {\left (6 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^2}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a d^4}-\frac {\left (6 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-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}(2,x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}-\frac {\left (6 b f^3\right ) \int \sinh (c+d x) \, dx}{a^2 d^3} \\ & = -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {6 b f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4}+\frac {\left (3 b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right ) \, dx}{2 a^3 d^3}-\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^3}-\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d^3} \\ & = -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {6 b f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4}+\frac {\left (3 b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a^3 d^4}-\frac {\left (6 \left (a^2+b^2\right ) 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 d^4}-\frac {\left (6 \left (a^2+b^2\right ) 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 d^4} \\ & = -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {6 b f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}-\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4}+\frac {3 b^2 f^3 \operatorname {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^3 d^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(3868\) vs. \(2(972)=1944\).
Time = 12.73 (sec) , antiderivative size = 3868, normalized size of antiderivative = 3.98 \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{3} \coth \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 13683 vs. \(2 (908) = 1816\).
Time = 0.48 (sec) , antiderivative size = 13683, normalized size of antiderivative = 14.08 \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \coth ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \coth \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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