Integrand size = 8, antiderivative size = 19 \[ \int e^x \sinh (4 x) \, dx=\frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \]
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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} \[ \int e^x \sinh (4 x) \, dx=\frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \]
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Rule 12
Rule 14
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-1+x^8}{2 x^4} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-1+x^8}{x^4} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x^4}+x^4\right ) \, dx,x,e^x\right ) \\ & = \frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int e^x \sinh (4 x) \, dx=\frac {e^{-3 x}}{6}+\frac {e^{5 x}}{10} \]
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Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {{\mathrm e}^{5 x}}{10}+\frac {{\mathrm e}^{-3 x}}{6}\) | \(14\) |
parallelrisch | \(\frac {{\mathrm e}^{x} \left (4 \cosh \left (4 x \right )-\sinh \left (4 x \right )\right )}{15}\) | \(18\) |
default | \(-\frac {\sinh \left (3 x \right )}{6}+\frac {\sinh \left (5 x \right )}{10}+\frac {\cosh \left (3 x \right )}{6}+\frac {\cosh \left (5 x \right )}{10}\) | \(26\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.42 \[ \int e^x \sinh (4 x) \, dx=\frac {4 \, {\left (\cosh \left (x\right )^{4} - \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}}{15 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^x \sinh (4 x) \, dx=- \frac {e^{x} \sinh {\left (4 x \right )}}{15} + \frac {4 e^{x} \cosh {\left (4 x \right )}}{15} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (4 x) \, dx=\frac {1}{10} \, e^{\left (5 \, x\right )} + \frac {1}{6} \, e^{\left (-3 \, x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (4 x) \, dx=\frac {1}{10} \, e^{\left (5 \, x\right )} + \frac {1}{6} \, e^{\left (-3 \, x\right )} \]
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Time = 1.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (4 x) \, dx=\frac {{\mathrm {e}}^{-3\,x}}{6}+\frac {{\mathrm {e}}^{5\,x}}{10} \]
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