Integrand size = 16, antiderivative size = 88 \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=-\frac {2 d^2 e^{a+b x}}{b \left (b^2-4 d^2\right )}+\frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \cosh (c+d x) \sinh (c+d x)}{b^2-4 d^2} \]
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Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5585, 2225} \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=\frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \sinh (c+d x) \cosh (c+d x)}{b^2-4 d^2}-\frac {2 d^2 e^{a+b x}}{b \left (b^2-4 d^2\right )} \]
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Rule 2225
Rule 5585
Rubi steps \begin{align*} \text {integral}& = \frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \cosh (c+d x) \sinh (c+d x)}{b^2-4 d^2}-\frac {\left (2 d^2\right ) \int e^{a+b x} \, dx}{b^2-4 d^2} \\ & = -\frac {2 d^2 e^{a+b x}}{b \left (b^2-4 d^2\right )}+\frac {b e^{a+b x} \cosh ^2(c+d x)}{b^2-4 d^2}-\frac {2 d e^{a+b x} \cosh (c+d x) \sinh (c+d x)}{b^2-4 d^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=\frac {e^{a+b x} \left (b^2-4 d^2+b^2 \cosh (2 (c+d x))-2 b d \sinh (2 (c+d x))\right )}{2 \left (b^3-4 b d^2\right )} \]
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Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{b x +2 d x +a +2 c}}{4 b +8 d}+\frac {{\mathrm e}^{b x -2 d x +a -2 c}}{4 b -8 d}\) | \(57\) |
parallelrisch | \(\frac {{\mathrm e}^{b x +a} \left (\cosh \left (2 d x +2 c \right ) b^{2}-2 b d \sinh \left (2 d x +2 c \right )+b^{2}-4 d^{2}\right )}{2 b^{3}-8 b \,d^{2}}\) | \(57\) |
default | \(\frac {\sinh \left (b x +a \right )}{2 b}+\frac {\sinh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\sinh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d}+\frac {\cosh \left (b x +a \right )}{2 b}+\frac {\cosh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\cosh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d}\) | \(112\) |
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Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.66 \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=\frac {b^{2} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} + {\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (b^{2} - 4 \, d^{2}\right )} \cosh \left (b x + a\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} - 4 \, d^{2}\right )} \sinh \left (b x + a\right ) - 4 \, {\left (b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) + b d \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (b^{3} - 4 \, b d^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (78) = 156\).
Time = 0.84 (sec) , antiderivative size = 432, normalized size of antiderivative = 4.91 \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=\begin {cases} x e^{a} \cosh ^{2}{\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\left (- \frac {x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac {x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d}\right ) e^{a} & \text {for}\: b = 0 \\\frac {x e^{a} e^{- 2 d x} \sinh ^{2}{\left (c + d x \right )}}{4} + \frac {x e^{a} e^{- 2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x e^{a} e^{- 2 d x} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {e^{a} e^{- 2 d x} \sinh ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 e^{a} e^{- 2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} & \text {for}\: b = - 2 d \\\frac {x e^{a} e^{2 d x} \sinh ^{2}{\left (c + d x \right )}}{4} - \frac {x e^{a} e^{2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x e^{a} e^{2 d x} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {e^{a} e^{2 d x} \sinh ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 e^{a} e^{2 d x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} & \text {for}\: b = 2 d \\\frac {b^{2} e^{a} e^{b x} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 b d e^{a} e^{b x} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{3} - 4 b d^{2}} + \frac {2 d^{2} e^{a} e^{b x} \sinh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 d^{2} e^{a} e^{b x} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=\frac {e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{4 \, {\left (b + 2 \, d\right )}} + \frac {e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{4 \, {\left (b - 2 \, d\right )}} + \frac {e^{\left (b x + a\right )}}{2 \, b} \]
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Time = 2.54 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.77 \[ \int e^{a+b x} \cosh ^2(c+d x) \, dx=\frac {2\,d^2\,{\mathrm {e}}^{a+b\,x}-b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {e}}^{a+b\,x}+2\,b\,d\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {e}}^{a+b\,x}\,\mathrm {sinh}\left (c+d\,x\right )}{4\,b\,d^2-b^3} \]
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