Integrand size = 26, antiderivative size = 631 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=-\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \]
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Time = 0.74 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5572, 3405, 3403, 2296, 2221, 2611, 2320, 6724, 5680, 2317, 2438} \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )}-\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )^{3/2}}+\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )^{3/2}}+\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}-\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}+\frac {3 f^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d^3 \left (a^2+b^2\right )}+\frac {3 f^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d^3 \left (a^2+b^2\right )}+\frac {3 a f (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{2 b d^2 \left (a^2+b^2\right )^{3/2}}-\frac {3 a f (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{2 b d^2 \left (a^2+b^2\right )^{3/2}}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac {3 f (e+f x)^2}{2 b d^2 \left (a^2+b^2\right )}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3403
Rule 3405
Rule 5572
Rule 5680
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}+\frac {(3 f) \int \frac {(e+f x)^2}{(a+b \sinh (c+d x))^2} \, dx}{2 b d} \\ & = -\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac {(3 a f) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{2 b \left (a^2+b^2\right ) d}+\frac {\left (3 f^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right ) d^2} \\ & = -\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac {(3 a f) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b \left (a^2+b^2\right ) d}+\frac {\left (3 f^2\right ) \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right ) d^2}+\frac {\left (3 f^2\right ) \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right ) d^2} \\ & = -\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}+\frac {(3 a f) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2} d}-\frac {(3 a f) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2} d}-\frac {\left (3 f^3\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right ) d^3}-\frac {\left (3 f^3\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right ) d^3} \\ & = -\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {\left (3 a f^2\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (3 a f^2\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^4} \\ & = -\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {\left (3 a f^3\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (3 a f^3\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b \left (a^2+b^2\right )^{3/2} d^3} \\ & = -\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {\left (3 a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {\left (3 a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right )^{3/2} d^4} \\ & = -\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(5753\) vs. \(2(631)=1262\).
Time = 13.70 (sec) , antiderivative size = 5753, normalized size of antiderivative = 9.12 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )}{\left (a +b \sinh \left (d x +c \right )\right )^{3}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 11757 vs. \(2 (575) = 1150\).
Time = 0.43 (sec) , antiderivative size = 11757, normalized size of antiderivative = 18.63 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \]
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