Integrand size = 8, antiderivative size = 25 \[ \int \cosh ^2(a+b x) \, dx=\frac {x}{2}+\frac {\cosh (a+b x) \sinh (a+b x)}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 8} \[ \int \cosh ^2(a+b x) \, dx=\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2} \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {\int 1 \, dx}{2} \\ & = \frac {x}{2}+\frac {\cosh (a+b x) \sinh (a+b x)}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \cosh ^2(a+b x) \, dx=\frac {2 (a+b x)+\sinh (2 (a+b x))}{4 b} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {2 b x +\sinh \left (2 b x +2 a \right )}{4 b}\) | \(20\) |
derivativedivides | \(\frac {\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}}{b}\) | \(27\) |
default | \(\frac {\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}}{b}\) | \(27\) |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{2 b x +2 a}}{8 b}-\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}\) | \(33\) |
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \cosh ^2(a+b x) \, dx=\frac {b x + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \cosh ^2(a+b x) \, dx=\begin {cases} - \frac {x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {\sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \cosh ^2(a+b x) \, dx=\frac {1}{2} \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \cosh ^2(a+b x) \, dx=\frac {1}{2} \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \]
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Time = 1.54 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \cosh ^2(a+b x) \, dx=\frac {x}{2}+\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b} \]
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