Integrand size = 14, antiderivative size = 54 \[ \int e^{a+b x} \cosh (c+d x) \, dx=\frac {b e^{a+b x} \cosh (c+d x)}{b^2-d^2}-\frac {d e^{a+b x} \sinh (c+d x)}{b^2-d^2} \]
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Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5583} \[ \int e^{a+b x} \cosh (c+d x) \, dx=\frac {b e^{a+b x} \cosh (c+d x)}{b^2-d^2}-\frac {d e^{a+b x} \sinh (c+d x)}{b^2-d^2} \]
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Rule 5583
Rubi steps \begin{align*} \text {integral}& = \frac {b e^{a+b x} \cosh (c+d x)}{b^2-d^2}-\frac {d e^{a+b x} \sinh (c+d x)}{b^2-d^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int e^{a+b x} \cosh (c+d x) \, dx=\frac {e^{a+b x} (b \cosh (c+d x)-d \sinh (c+d x))}{(b-d) (b+d)} \]
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Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{b x +a} \left (b \cosh \left (d x +c \right )-d \sinh \left (d x +c \right )\right )}{b^{2}-d^{2}}\) | \(37\) |
risch | \(\frac {{\mathrm e}^{b x +d x +a +c}}{2 b +2 d}+\frac {{\mathrm e}^{b x -d x +a -c}}{2 b -2 d}\) | \(41\) |
default | \(\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{2 b -2 d}+\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d}+\frac {\cosh \left (a -c +\left (b -d \right ) x \right )}{2 b -2 d}+\frac {\cosh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d}\) | \(78\) |
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Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int e^{a+b x} \cosh (c+d x) \, dx=\frac {b \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + b \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - {\left (d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{b^{2} - d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (42) = 84\).
Time = 0.38 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.09 \[ \int e^{a+b x} \cosh (c+d x) \, dx=\begin {cases} x e^{a} \cosh {\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x e^{a} e^{- d x} \sinh {\left (c + d x \right )}}{2} + \frac {x e^{a} e^{- d x} \cosh {\left (c + d x \right )}}{2} + \frac {e^{a} e^{- d x} \sinh {\left (c + d x \right )}}{2 d} & \text {for}\: b = - d \\- \frac {x e^{a} e^{d x} \sinh {\left (c + d x \right )}}{2} + \frac {x e^{a} e^{d x} \cosh {\left (c + d x \right )}}{2} + \frac {e^{a} e^{d x} \sinh {\left (c + d x \right )}}{2 d} & \text {for}\: b = d \\\frac {b e^{a} e^{b x} \cosh {\left (c + d x \right )}}{b^{2} - d^{2}} - \frac {d e^{a} e^{b x} \sinh {\left (c + d x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int e^{a+b x} \cosh (c+d x) \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.74 \[ \int e^{a+b x} \cosh (c+d x) \, dx=\frac {e^{\left (b x + d x + a + c\right )}}{2 \, {\left (b + d\right )}} + \frac {e^{\left (b x - d x + a - c\right )}}{2 \, {\left (b - d\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int e^{a+b x} \cosh (c+d x) \, dx=\frac {{\mathrm {e}}^{a-c+b\,x-d\,x}\,\left (b+d+b\,{\mathrm {e}}^{2\,c+2\,d\,x}-d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{2\,\left (b^2-d^2\right )} \]
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