Integrand size = 8, antiderivative size = 19 \[ \int e^x \sinh (2 x) \, dx=\frac {e^{-x}}{2}+\frac {e^{3 x}}{6} \]
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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} \[ \int e^x \sinh (2 x) \, dx=\frac {e^{-x}}{2}+\frac {e^{3 x}}{6} \]
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Rule 12
Rule 14
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-1+x^4}{2 x^2} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-1+x^4}{x^2} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x^2}+x^2\right ) \, dx,x,e^x\right ) \\ & = \frac {e^{-x}}{2}+\frac {e^{3 x}}{6} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int e^x \sinh (2 x) \, dx=\frac {1}{6} e^{-x} \left (3+e^{4 x}\right ) \]
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Time = 0.35 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{6}+\frac {{\mathrm e}^{-x}}{2}\) | \(14\) |
parallelrisch | \(\frac {{\mathrm e}^{x} \left (2 \cosh \left (2 x \right )-\sinh \left (2 x \right )\right )}{3}\) | \(18\) |
default | \(-\frac {\sinh \left (x \right )}{2}+\frac {\sinh \left (3 x \right )}{6}+\frac {\cosh \left (x \right )}{2}+\frac {\cosh \left (3 x \right )}{6}\) | \(22\) |
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int e^x \sinh (2 x) \, dx=\frac {2 \, {\left (\cosh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}}{3 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^x \sinh (2 x) \, dx=- \frac {e^{x} \sinh {\left (2 x \right )}}{3} + \frac {2 e^{x} \cosh {\left (2 x \right )}}{3} \]
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Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (2 x) \, dx=\frac {1}{6} \, e^{\left (3 \, x\right )} + \frac {1}{2} \, e^{\left (-x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int e^x \sinh (2 x) \, dx=\frac {1}{6} \, e^{\left (3 \, x\right )} + \frac {1}{2} \, e^{\left (-x\right )} \]
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Time = 1.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int e^x \sinh (2 x) \, dx=\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{4\,x}+3\right )}{6} \]
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