Integrand size = 34, antiderivative size = 696 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d} \]
[Out]
Time = 0.76 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5698, 3392, 32, 3391, 5684, 3377, 2717, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {6 a f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {3 a f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}+\frac {3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 b d}+\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {(e+f x)^4}{8 b f} \]
[In]
[Out]
Rule 32
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 2717
Rule 3377
Rule 3391
Rule 3392
Rule 3403
Rule 5684
Rule 5698
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \cosh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {a^2 \int (e+f x)^3 \, dx}{b^3}-\frac {a \int (e+f x)^3 \sinh (c+d x) \, dx}{b^2}+\frac {\int (e+f x)^3 \, dx}{2 b}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {\left (3 f^2\right ) \int (e+f x) \cosh ^2(c+d x) \, dx}{2 b d^2} \\ & = \frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (2 a \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}{b^3}+\frac {(3 a f) \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2 d}+\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2} \\ & = \frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (2 a \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}{b^2}+\frac {\left (2 a \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}{b^2}-\frac {\left (6 a f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d^2} \\ & = \frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \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 )}{b^3 d}+\frac {a \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 )}{b^3 d}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {\left (3 a \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}{b^3 d}-\frac {\left (3 a \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}{b^3 d}+\frac {\left (6 a f^3\right ) \int \cosh (c+d x) \, dx}{b^2 d^3} \\ & = \frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \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 )}{b^3 d}+\frac {a \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 )}{b^3 d}-\frac {3 a \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 )}{b^3 d^2}+\frac {3 a \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 )}{b^3 d^2}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {\left (6 a \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}{b^3 d^2}-\frac {\left (6 a \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}{b^3 d^2} \\ & = \frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \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 )}{b^3 d}+\frac {a \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 )}{b^3 d}-\frac {3 a \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 )}{b^3 d^2}+\frac {3 a \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 )}{b^3 d^2}+\frac {6 a \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 )}{b^3 d^3}-\frac {6 a \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 )}{b^3 d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (6 a \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}{b^3 d^3}+\frac {\left (6 a \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}{b^3 d^3} \\ & = \frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \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 )}{b^3 d}+\frac {a \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 )}{b^3 d}-\frac {3 a \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 )}{b^3 d^2}+\frac {3 a \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 )}{b^3 d^2}+\frac {6 a \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 )}{b^3 d^3}-\frac {6 a \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 )}{b^3 d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (6 a \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 )}{b^3 d^4}+\frac {\left (6 a \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 )}{b^3 d^4} \\ & = \frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \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 )}{b^3 d}+\frac {a \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 )}{b^3 d}-\frac {3 a \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 )}{b^3 d^2}+\frac {3 a \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 )}{b^3 d^2}+\frac {6 a \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 )}{b^3 d^3}-\frac {6 a \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 )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1994\) vs. \(2(696)=1392\).
Time = 9.61 (sec) , antiderivative size = 1994, normalized size of antiderivative = 2.86 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e^3 \left (\frac {c}{d}+x-\frac {2 a \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}\right )}{4 b}+\frac {3 e^2 f \left (x^2-\frac {2 a \left (d x \left (\log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^2}\right )}{8 b}+\frac {e f^2 \left (x^3-\frac {3 a \left (d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}\right )}{4 b}+\frac {f^3 \left (x^4-\frac {4 a \left (d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-6 d x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+6 d x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^4}\right )}{16 b}+\frac {e f^2 \left (2 \left (4 a^2+b^2\right ) x^3-\frac {6 a \left (4 a^2+3 b^2\right ) \left (d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}-\frac {24 a b \cosh (d x) \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right )}{d^3}+\frac {3 b^2 \cosh (2 d x) \left (-2 d x \cosh (2 c)+\left (1+2 d^2 x^2\right ) \sinh (2 c)\right )}{d^3}-\frac {24 a b \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}+\frac {3 b^2 \left (\left (1+2 d^2 x^2\right ) \cosh (2 c)-2 d x \sinh (2 c)\right ) \sinh (2 d x)}{d^3}\right )}{8 b^3}+\frac {f^3 \left (\left (4 a^2+b^2\right ) x^4-\frac {4 a \left (4 a^2+3 b^2\right ) \left (d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-d^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-6 d x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+6 d x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^4}-\frac {16 a b \cosh (d x) \left (d x \left (6+d^2 x^2\right ) \cosh (c)-3 \left (2+d^2 x^2\right ) \sinh (c)\right )}{d^4}+\frac {b^2 \cosh (2 d x) \left (-3 \left (1+2 d^2 x^2\right ) \cosh (2 c)+2 d x \left (3+2 d^2 x^2\right ) \sinh (2 c)\right )}{d^4}-\frac {16 a b \left (-3 \left (2+d^2 x^2\right ) \cosh (c)+d x \left (6+d^2 x^2\right ) \sinh (c)\right ) \sinh (d x)}{d^4}+\frac {b^2 \left (2 d x \left (3+2 d^2 x^2\right ) \cosh (2 c)-3 \left (1+2 d^2 x^2\right ) \sinh (2 c)\right ) \sinh (2 d x)}{d^4}\right )}{16 b^3}+\frac {e^3 \left (\left (4 a^2+b^2\right ) (c+d x)-\frac {2 a \left (4 a^2+3 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \cosh (c+d x)+b^2 \sinh (2 (c+d x))\right )}{4 b^3 d}+\frac {3 e^2 f \left (\left (4 a^2+b^2\right ) (-c+d x) (c+d x)-8 a b d x \cosh (c+d x)-b^2 \cosh (2 (c+d x))-\frac {2 a \left (4 a^2+3 b^2\right ) \left (2 c \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-(c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+8 a b \sinh (c+d x)+2 b^2 d x \sinh (2 (c+d x))\right )}{8 b^3 d^2} \]
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\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 3847 vs. \(2 (638) = 1276\).
Time = 0.34 (sec) , antiderivative size = 3847, normalized size of antiderivative = 5.53 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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