Integrand size = 16, antiderivative size = 49 \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=-\frac {e^{-a-b x}}{4 b}-\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b} \]
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Time = 0.02 (sec) , antiderivative size = 49, 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} \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=-\frac {e^{-a-b x}}{4 b}-\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b} \]
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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 )^2}{4 x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2} \, dx,x,e^{a+b x}\right )}{4 b} \\ & = \frac {\text {Subst}\left (\int \left (-2+\frac {1}{x^2}+x^2\right ) \, dx,x,e^{a+b x}\right )}{4 b} \\ & = -\frac {e^{-a-b x}}{4 b}-\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{12 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=\frac {e^{-a-b x} \left (-3-6 e^{2 (a+b x)}+e^{4 (a+b x)}\right )}{12 b} \]
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Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )+\frac {\sinh \left (b x +a \right )^{3}}{3}}{b}\) | \(35\) |
default | \(\frac {\left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )+\frac {\sinh \left (b x +a \right )^{3}}{3}}{b}\) | \(35\) |
risch | \(-\frac {{\mathrm e}^{-b x -a}}{4 b}-\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{3 b x +3 a}}{12 b}\) | \(41\) |
parallelrisch | \(-\frac {{\mathrm e}^{b x +a} \left (-2 \sinh \left (2 b x +2 a \right )-4 \sinh \left (b x +a \right )+\cosh \left (2 b x +2 a \right )+4 \cosh \left (b x +a \right )+3\right )}{6 b}\) | \(50\) |
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Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10 \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=-\frac {\cosh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 3}{6 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (34) = 68\).
Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.59 \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=\begin {cases} \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b} - \frac {2 e^{a} e^{b x} \cosh ^{2}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )}}{12 \, b} - \frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{4 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.69 \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )} - 6 \, e^{\left (b x + a\right )} - 3 \, e^{\left (-b x - a\right )}}{12 \, b} \]
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Time = 1.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int e^{a+b x} \sinh ^2(a+b x) \, dx=-\frac {6\,{\mathrm {e}}^{a+b\,x}+3\,{\mathrm {e}}^{-a-b\,x}-{\mathrm {e}}^{3\,a+3\,b\,x}}{12\,b} \]
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