Integrand size = 16, antiderivative size = 32 \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=-\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{6 b} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2320, 12} \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=\frac {e^{3 a+3 b x}}{6 b}-\frac {e^{a+b x}}{2 b} \]
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Rule 12
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{2} \left (-1+x^2\right ) \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,e^{a+b x}\right )}{2 b} \\ & = -\frac {e^{a+b x}}{2 b}+\frac {e^{3 a+3 b x}}{6 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=\frac {e^{a+b x} \left (-3+e^{2 (a+b x)}\right )}{6 b} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{3 b x +3 a}}{6 b}\) | \(27\) |
parallelrisch | \(-\frac {{\mathrm e}^{2 b x +2 a} \left (\cosh \left (b x +a \right )-2 \sinh \left (b x +a \right )\right )}{3 b}\) | \(30\) |
default | \(-\frac {\sinh \left (b x +a \right )}{2 b}+\frac {\sinh \left (3 b x +3 a \right )}{6 b}-\frac {\cosh \left (b x +a \right )}{2 b}+\frac {\cosh \left (3 b x +3 a \right )}{6 b}\) | \(52\) |
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=\frac {\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 3}{6 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=\begin {cases} \frac {2 e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )}}{3 b} - \frac {e^{2 a} e^{2 b x} \cosh {\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh {\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )}}{6 \, b} - \frac {e^{\left (b x + a\right )}}{2 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )}}{6 \, b} - \frac {e^{\left (b x + a\right )}}{2 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int e^{2 (a+b x)} \sinh (a+b x) \, dx=-\frac {3\,{\mathrm {e}}^{a+b\,x}-{\mathrm {e}}^{3\,a+3\,b\,x}}{6\,b} \]
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