Integrand size = 26, antiderivative size = 448 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d} \]
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Time = 0.53 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5713, 5698, 3377, 2718, 5680, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^4}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b (e+f x)^3 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d}+\frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^3 \cosh (c+d x)}{a d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {(e+f x)^3 \sinh (c+d x)}{a d} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 2718
Rule 3377
Rule 5680
Rule 5698
Rule 5713
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx \\ & = \frac {\int (e+f x)^3 \cosh (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx}{a} \\ & = \frac {b (e+f x)^4}{4 a^2 f}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a}-\frac {(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d} \\ & = \frac {b (e+f x)^4}{4 a^2 f}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d}+\frac {\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2} \\ & = \frac {b (e+f x)^4}{4 a^2 f}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {\left (6 b 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^2 d^2}+\frac {\left (6 b 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^2 d^2}-\frac {\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3} \\ & = \frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^3}-\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 d^3} \\ & = \frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {\left (6 b 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^2 d^4}-\frac {\left (6 b 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^2 d^4} \\ & = \frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1792\) vs. \(2(448)=896\).
Time = 20.20 (sec) , antiderivative size = 1792, normalized size of antiderivative = 4.00 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {e f^2 \text {csch}(c+d x) \left (\frac {4 b x^3}{-1+e^{2 c}}-2 b x^3 \coth (c)-\frac {6 a^2 b \left (d^2 x^2 \log \left (1+\frac {\left (b-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 d x \operatorname {PolyLog}\left (2,\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 \operatorname {PolyLog}\left (3,\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )\right )}{\sqrt {a^2+b^2} \left (-b+\sqrt {a^2+b^2}\right ) d^3}-\frac {6 a^2 b \left (d^2 x^2 \log \left (1+\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-2 \operatorname {PolyLog}\left (3,-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )\right )}{\sqrt {a^2+b^2} \left (b+\sqrt {a^2+b^2}\right ) d^3}+\frac {6 b^2 \left (d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}-\frac {6 b^2 \left (d^2 x^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}+\frac {6 a \cosh (d x) \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right )}{d^3}+\frac {6 a \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right ) \sinh (d x)}{d^3}\right ) (b+a \sinh (c+d x))}{2 a^2 (a+b \text {csch}(c+d x))}+\frac {f^3 \text {csch}(c+d x) \left (\frac {b \left (x^4-\frac {2 a^2 \left (-1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {\left (b-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-6 d x \operatorname {PolyLog}\left (3,\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-6 \operatorname {PolyLog}\left (4,\frac {\left (-b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )\right )}{\sqrt {a^2+b^2} \left (-b+\sqrt {a^2+b^2}\right ) d^4}-\frac {2 a^2 \left (-1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-3 d^2 x^2 \operatorname {PolyLog}\left (2,-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-6 d x \operatorname {PolyLog}\left (3,-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )-6 \operatorname {PolyLog}\left (4,-\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{a}\right )\right )}{\sqrt {a^2+b^2} \left (b+\sqrt {a^2+b^2}\right ) d^4}+\frac {2 b \left (-1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )-6 d x \operatorname {PolyLog}\left (3,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^4}-\frac {2 b \left (-1+e^{2 c}\right ) \left (d^3 x^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+3 d^2 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-6 d x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+6 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^4}\right )}{a^2 \left (-1+e^{2 c}\right )}-\frac {b x^4 \cosh (c) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right )}{4 a^2}+\frac {2 \cosh (d x) \left (-6 \cosh (c)-3 d^2 x^2 \cosh (c)+6 d x \sinh (c)+d^3 x^3 \sinh (c)\right )}{a d^4}+\frac {2 \left (6 d x \cosh (c)+d^3 x^3 \cosh (c)-6 \sinh (c)-3 d^2 x^2 \sinh (c)\right ) \sinh (d x)}{a d^4}\right ) (b+a \sinh (c+d x))}{2 (a+b \text {csch}(c+d x))}-\frac {e^3 \text {csch}(c+d x) \left (\frac {b \log (b+a \sinh (c+d x))}{a^2}-\frac {\sinh (c+d x)}{a}\right ) (b+a \sinh (c+d x))}{d (a+b \text {csch}(c+d x))}+\frac {3 e^2 f \text {csch}(c+d x) (b+a \sinh (c+d x)) \left (-2 a \cosh (c+d x)-b \left (2 c (c+d x)-(c+d x)^2+2 (c+d x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+2 (c+d x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-2 c \log \left (a-2 b e^{c+d x}-a e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )+2 a d x \sinh (c+d x)\right )}{2 a^2 d^2 (a+b \text {csch}(c+d x))} \]
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\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )}{a +b \,\operatorname {csch}\left (d x +c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 1976 vs. \(2 (420) = 840\).
Time = 0.30 (sec) , antiderivative size = 1976, normalized size of antiderivative = 4.41 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cosh {\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]
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