Integrand size = 19, antiderivative size = 26 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\frac {(a \cosh (c+d x)+a \sinh (c+d x))^2}{2 d} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3150} \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\frac {(a \sinh (c+d x)+a \cosh (c+d x))^2}{2 d} \]
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Rule 3150
Rubi steps \begin{align*} \text {integral}& = \frac {(a \cosh (c+d x)+a \sinh (c+d x))^2}{2 d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\frac {a^2 (\cosh (c+d x)+\sinh (c+d x))^2}{2 d} \]
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Time = 1.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {a^{2} {\mathrm e}^{2 d x +2 c}}{2 d}\) | \(18\) |
gosper | \(\frac {a^{2} \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )^{2}}{2 d}\) | \(24\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} \cosh \left (d x +c \right )^{2}+a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}\) | \(70\) |
default | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} \cosh \left (d x +c \right )^{2}+a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}\) | \(70\) |
parts | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}+\frac {a^{2} \cosh \left (d x +c \right )^{2}}{d}\) | \(75\) |
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none
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\frac {a^{2} \cosh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right ) - d \sinh \left (d x + c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )}}{d} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sinh {\left (c \right )} + a \cosh {\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.38 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\frac {1}{8} \, a^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac {a^{2} \cosh \left (d x + c\right )^{2}}{d} \]
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\frac {a^{2} e^{\left (2 \, d x + 2 \, c\right )}}{2 \, d} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{2\,d} \]
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