Integrand size = 16, antiderivative size = 83 \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=-\frac {e^{-3 a-3 b x}}{48 b}+\frac {e^{-a-b x}}{4 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{12 b}+\frac {e^{5 a+5 b x}}{80 b} \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2320, 12, 276} \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=-\frac {e^{-3 a-3 b x}}{48 b}+\frac {e^{-a-b x}}{4 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{12 b}+\frac {e^{5 a+5 b x}}{80 b} \]
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Rule 12
Rule 276
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{16 x^4} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^4} \, dx,x,e^{a+b x}\right )}{16 b} \\ & = \frac {\text {Subst}\left (\int \left (6+\frac {1}{x^4}-\frac {4}{x^2}-4 x^2+x^4\right ) \, dx,x,e^{a+b x}\right )}{16 b} \\ & = -\frac {e^{-3 a-3 b x}}{48 b}+\frac {e^{-a-b x}}{4 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{12 b}+\frac {e^{5 a+5 b x}}{80 b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75 \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=\frac {e^{-3 (a+b x)} \left (-5+60 e^{2 (a+b x)}+90 e^{4 (a+b x)}-20 e^{6 (a+b x)}+3 e^{8 (a+b x)}\right )}{240 b} \]
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Time = 1.64 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(\frac {\left (\frac {8}{15}+\frac {\sinh \left (b x +a \right )^{4}}{5}-\frac {4 \sinh \left (b x +a \right )^{2}}{15}\right ) \cosh \left (b x +a \right )+\frac {\sinh \left (b x +a \right )^{5}}{5}}{b}\) | \(45\) |
default | \(\frac {\left (\frac {8}{15}+\frac {\sinh \left (b x +a \right )^{4}}{5}-\frac {4 \sinh \left (b x +a \right )^{2}}{15}\right ) \cosh \left (b x +a \right )+\frac {\sinh \left (b x +a \right )^{5}}{5}}{b}\) | \(45\) |
risch | \(-\frac {{\mathrm e}^{-3 b x -3 a}}{48 b}+\frac {{\mathrm e}^{-b x -a}}{4 b}+\frac {3 \,{\mathrm e}^{b x +a}}{8 b}-\frac {{\mathrm e}^{3 b x +3 a}}{12 b}+\frac {{\mathrm e}^{5 b x +5 a}}{80 b}\) | \(69\) |
parallelrisch | \(\frac {{\mathrm e}^{b x +a} \left (-\cosh \left (4 b x +4 a \right )+64 \cosh \left (b x +a \right )+4 \sinh \left (4 b x +4 a \right )-64 \sinh \left (b x +a \right )-40 \sinh \left (2 b x +2 a \right )+20 \cosh \left (2 b x +2 a \right )+45\right )}{120 b}\) | \(74\) |
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36 \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=-\frac {\cosh \left (b x + a\right )^{4} - 16 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 10\right )} \sinh \left (b x + a\right )^{2} - 20 \, \cosh \left (b x + a\right )^{2} - 16 \, {\left (\cosh \left (b x + a\right )^{3} - 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 45}{120 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (65) = 130\).
Time = 2.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.67 \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=\begin {cases} \frac {e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )}}{5 b} + \frac {4 e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{5 b} - \frac {4 e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{5 b} - \frac {8 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{15 b} + \frac {8 e^{a} e^{b x} \cosh ^{4}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.82 \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=\frac {e^{\left (5 \, b x + 5 \, a\right )}}{80 \, b} - \frac {e^{\left (3 \, b x + 3 \, a\right )}}{12 \, b} + \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} + \frac {e^{\left (-b x - a\right )}}{4 \, b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=\frac {5 \, {\left (12 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (5 \, b x + 5 \, a\right )} - 20 \, e^{\left (3 \, b x + 3 \, a\right )} + 90 \, e^{\left (b x + a\right )}}{240 \, b} \]
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Time = 0.53 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70 \[ \int e^{a+b x} \sinh ^4(a+b x) \, dx=\frac {90\,{\mathrm {e}}^{a+b\,x}+60\,{\mathrm {e}}^{-a-b\,x}-5\,{\mathrm {e}}^{-3\,a-3\,b\,x}-20\,{\mathrm {e}}^{3\,a+3\,b\,x}+3\,{\mathrm {e}}^{5\,a+5\,b\,x}}{240\,b} \]
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