Integrand size = 8, antiderivative size = 19 \[ \int e^x \sinh (3 x) \, dx=\frac {e^{-2 x}}{4}+\frac {e^{4 x}}{8} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2320, 12, 14} \[ \int e^x \sinh (3 x) \, dx=\frac {e^{-2 x}}{4}+\frac {e^{4 x}}{8} \]
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Rule 12
Rule 14
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-1+x^6}{2 x^3} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-1+x^6}{x^3} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x^3}+x^3\right ) \, dx,x,e^x\right ) \\ & = \frac {e^{-2 x}}{4}+\frac {e^{4 x}}{8} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int e^x \sinh (3 x) \, dx=\frac {1}{8} e^{-2 x} \left (2+e^{6 x}\right ) \]
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Time = 0.64 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {{\mathrm e}^{4 x}}{8}+\frac {{\mathrm e}^{-2 x}}{4}\) | \(14\) |
parallelrisch | \(-\frac {{\mathrm e}^{x} \left (-3 \cosh \left (3 x \right )+\sinh \left (3 x \right )\right )}{8}\) | \(16\) |
default | \(-\frac {\sinh \left (2 x \right )}{4}+\frac {\sinh \left (4 x \right )}{8}+\frac {\cosh \left (2 x \right )}{4}+\frac {\cosh \left (4 x \right )}{8}\) | \(26\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int e^x \sinh (3 x) \, dx=\frac {3 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 9 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - \sinh \left (x\right )^{3}}{8 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^x \sinh (3 x) \, dx=- \frac {e^{x} \sinh {\left (3 x \right )}}{8} + \frac {3 e^{x} \cosh {\left (3 x \right )}}{8} \]
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none
Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (3 x) \, dx=\frac {1}{8} \, e^{\left (4 \, x\right )} + \frac {1}{4} \, e^{\left (-2 \, x\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (3 x) \, dx=\frac {1}{8} \, e^{\left (4 \, x\right )} + \frac {1}{4} \, e^{\left (-2 \, x\right )} \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int e^x \sinh (3 x) \, dx=\frac {{\mathrm {e}}^{-2\,x}\,\left ({\mathrm {e}}^{6\,x}+2\right )}{8} \]
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