Integrand size = 26, antiderivative size = 335 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2} \]
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Time = 0.43 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5676, 3391, 3377, 2717, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 e x}{b^3}+\frac {a^2 f x^2}{2 b^3}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}-\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a f \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {e x}{2 b}-\frac {f x^2}{4 b} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2717
Rule 3377
Rule 3391
Rule 3403
Rule 5676
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = \frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}-\frac {a \int (e+f x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {\int (e+f x) \, dx}{2 b} \\ & = -\frac {e x}{2 b}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {a^2 \int (e+f x) \, dx}{b^3}-\frac {a^3 \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(a f) \int \cosh (c+d x) \, dx}{b^2 d} \\ & = \frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^3} \\ & = \frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2 \sqrt {a^2+b^2}}+\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2 \sqrt {a^2+b^2}} \\ & = \frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (a^3 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d}-\frac {\left (a^3 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d} \\ & = \frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (a^3 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 \sqrt {a^2+b^2} d^2}-\frac {\left (a^3 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 \sqrt {a^2+b^2} d^2} \\ & = \frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2} \\ \end{align*}
Time = 5.51 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.91 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 \left (2 a^2-b^2\right ) (c+d x) (c f-d (2 e+f x))+8 a b d (e+f x) \cosh (c+d x)+b^2 f \cosh (2 (c+d x))+\frac {8 a^3 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-8 a b f \sinh (c+d x)-2 b^2 d (e+f x) \sinh (2 (c+d x))}{8 b^3 d^2} \]
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Time = 1.73 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.76
method | result | size |
risch | \(\frac {a^{2} f \,x^{2}}{2 b^{3}}-\frac {f \,x^{2}}{4 b}+\frac {a^{2} e x}{b^{3}}-\frac {e x}{2 b}+\frac {\left (2 d f x +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 b \,d^{2}}-\frac {a \left (d f x +d e -f \right ) {\mathrm e}^{d x +c}}{2 b^{2} d^{2}}-\frac {a \left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b^{2} d^{2}}-\frac {\left (2 d f x +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 b \,d^{2}}+\frac {2 a^{3} e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}-\frac {2 a^{3} f c \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}\) | \(589\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1727 vs. \(2 (303) = 606\).
Time = 0.31 (sec) , antiderivative size = 1727, normalized size of antiderivative = 5.16 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
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