Integrand size = 28, antiderivative size = 864 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^4}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d} \]
[Out]
Time = 0.83 (sec) , antiderivative size = 864, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {5713, 5698, 3392, 3377, 2718, 3391, 5684, 5554, 32, 2715, 8, 5680, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {b \sinh ^2(c+d x) (e+f x)^3}{2 a^2 d}-\frac {b \left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}-\frac {b \left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}+\frac {\cosh ^2(c+d x) \sinh (c+d x) (e+f x)^3}{3 a d}+\frac {b^2 \sinh (c+d x) (e+f x)^3}{a^3 d}+\frac {2 \sinh (c+d x) (e+f x)^3}{3 a d}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {f \cosh ^3(c+d x) (e+f x)^2}{3 a d^2}-\frac {3 b^2 f \cosh (c+d x) (e+f x)^2}{a^3 d^2}-\frac {2 f \cosh (c+d x) (e+f x)^2}{a d^2}-\frac {3 b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}+\frac {3 b f \cosh (c+d x) \sinh (c+d x) (e+f x)^2}{4 a^2 d^2}-\frac {3 b f^2 \sinh ^2(c+d x) (e+f x)}{4 a^2 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac {6 b^2 f^2 \sinh (c+d x) (e+f x)}{a^3 d^3}+\frac {40 f^2 \sinh (c+d x) (e+f x)}{9 a d^3}+\frac {2 f^2 \cosh ^2(c+d x) \sinh (c+d x) (e+f x)}{9 a d^3}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {3 b f^3 x}{8 a^2 d^3}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^4}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4} \]
[In]
[Out]
Rule 8
Rule 32
Rule 2221
Rule 2320
Rule 2611
Rule 2715
Rule 2718
Rule 3377
Rule 3391
Rule 3392
Rule 5554
Rule 5680
Rule 5684
Rule 5698
Rule 5713
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx \\ & = \frac {\int (e+f x)^3 \cosh ^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)} \, dx}{a} \\ & = -\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}+\frac {2 \int (e+f x)^3 \cosh (c+d x) \, dx}{3 a}-\frac {b \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \cosh (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)}{b+a \sinh (c+d x)} \, dx}{a^3}+\frac {\left (2 f^2\right ) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 a d^2} \\ & = \frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{b-\sqrt {a^2+b^2}+a 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}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac {(2 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}+\frac {(3 b f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 a^2 d}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{a^3 d}+\frac {\left (4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{9 a d^2} \\ & = \frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}+\frac {4 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {(3 b f) \int (e+f x)^2 \, dx}{4 a^2 d}+\frac {\left (3 b \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}+\frac {\left (3 b \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}+\frac {\left (4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\frac {\left (6 b^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a^3 d^2}-\frac {\left (4 f^3\right ) \int \sinh (c+d x) \, dx}{9 a d^3}+\frac {\left (3 b f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 a^2 d^3} \\ & = -\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {4 f^3 \cosh (c+d x)}{9 a d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}+\frac {\left (6 b \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^2}+\frac {\left (6 b \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^2}-\frac {\left (4 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}-\frac {\left (3 b f^3\right ) \int 1 \, dx}{8 a^2 d^3}-\frac {\left (6 b^2 f^3\right ) \int \sinh (c+d x) \, dx}{a^3 d^3} \\ & = -\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^3}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^3} \\ & = -\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^4}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^4} \\ & = -\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^4}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(7404\) vs. \(2(864)=1728\).
Time = 32.30 (sec) , antiderivative size = 7404, normalized size of antiderivative = 8.57 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{3}}{a +b \,\operatorname {csch}\left (d x +c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 7980 vs. \(2 (810) = 1620\).
Time = 0.38 (sec) , antiderivative size = 7980, normalized size of antiderivative = 9.24 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]
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