Integrand size = 18, antiderivative size = 66 \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=\frac {e^{-a-b x}}{8 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{8 b}+\frac {e^{5 a+5 b x}}{40 b} \]
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Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 12, 276} \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=\frac {e^{-a-b x}}{8 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{8 b}+\frac {e^{5 a+5 b x}}{40 b} \]
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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 )^3}{8 x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right )^3}{x^2} \, dx,x,e^{a+b x}\right )}{8 b} \\ & = \frac {\text {Subst}\left (\int \left (3-\frac {1}{x^2}-3 x^2+x^4\right ) \, dx,x,e^{a+b x}\right )}{8 b} \\ & = \frac {e^{-a-b x}}{8 b}+\frac {3 e^{a+b x}}{8 b}-\frac {e^{3 a+3 b x}}{8 b}+\frac {e^{5 a+5 b x}}{40 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=\frac {e^{-a-b x} \left (5+15 e^{2 (a+b x)}-5 e^{4 (a+b x)}+e^{6 (a+b x)}\right )}{40 b} \]
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Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {{\mathrm e}^{-b x -a}}{8 b}+\frac {3 \,{\mathrm e}^{b x +a}}{8 b}-\frac {{\mathrm e}^{3 b x +3 a}}{8 b}+\frac {{\mathrm e}^{5 b x +5 a}}{40 b}\) | \(55\) |
parallelrisch | \(-\frac {{\mathrm e}^{2 b x +2 a} \left (8 \sinh \left (2 b x +2 a \right )+2 \sinh \left (3 b x +3 a \right )+10 \sinh \left (b x +a \right )-8 \cosh \left (2 b x +2 a \right )-3 \cosh \left (3 b x +3 a \right )-5 \cosh \left (b x +a \right )\right )}{20 b}\) | \(76\) |
default | \(\frac {\sinh \left (b x +a \right )}{4 b}-\frac {\sinh \left (3 b x +3 a \right )}{8 b}+\frac {\sinh \left (5 b x +5 a \right )}{40 b}+\frac {\cosh \left (b x +a \right )}{2 b}-\frac {\cosh \left (3 b x +3 a \right )}{8 b}+\frac {\cosh \left (5 b x +5 a \right )}{40 b}\) | \(80\) |
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Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.59 \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=\frac {3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 2 \, \sinh \left (b x + a\right )^{3} - 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right ) + 5 \, \cosh \left (b x + a\right )}{20 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (49) = 98\).
Time = 0.88 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.88 \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=\begin {cases} \frac {2 e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )}}{5 b} + \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{5 b} - \frac {4 e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{5 b} + \frac {2 e^{2 a} e^{2 b x} \cosh ^{3}{\left (a + b x \right )}}{5 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=-\frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 15 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (5 \, b x + 5 \, a\right )}}{40 \, b} + \frac {e^{\left (-b x - a\right )}}{8 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=\frac {e^{\left (5 \, b x + 5 \, a\right )}}{40 \, b} - \frac {e^{\left (3 \, b x + 3 \, a\right )}}{8 \, b} + \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} + \frac {e^{\left (-b x - a\right )}}{8 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.68 \[ \int e^{2 (a+b x)} \sinh ^3(a+b x) \, dx=\frac {15\,{\mathrm {e}}^{a+b\,x}+5\,{\mathrm {e}}^{-a-b\,x}-5\,{\mathrm {e}}^{3\,a+3\,b\,x}+{\mathrm {e}}^{5\,a+5\,b\,x}}{40\,b} \]
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