Integrand size = 8, antiderivative size = 41 \[ \int \cosh ^5(a+b x) \, dx=\frac {\sinh (a+b x)}{b}+\frac {2 \sinh ^3(a+b x)}{3 b}+\frac {\sinh ^5(a+b x)}{5 b} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713} \[ \int \cosh ^5(a+b x) \, dx=\frac {\sinh ^5(a+b x)}{5 b}+\frac {2 \sinh ^3(a+b x)}{3 b}+\frac {\sinh (a+b x)}{b} \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-i \sinh (a+b x)\right )}{b} \\ & = \frac {\sinh (a+b x)}{b}+\frac {2 \sinh ^3(a+b x)}{3 b}+\frac {\sinh ^5(a+b x)}{5 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \cosh ^5(a+b x) \, dx=\frac {\sinh (a+b x)}{b}+\frac {2 \sinh ^3(a+b x)}{3 b}+\frac {\sinh ^5(a+b x)}{5 b} \]
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Time = 0.39 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\left (\frac {8}{15}+\frac {\cosh \left (b x +a \right )^{4}}{5}+\frac {4 \cosh \left (b x +a \right )^{2}}{15}\right ) \sinh \left (b x +a \right )}{b}\) | \(33\) |
default | \(\frac {\left (\frac {8}{15}+\frac {\cosh \left (b x +a \right )^{4}}{5}+\frac {4 \cosh \left (b x +a \right )^{2}}{15}\right ) \sinh \left (b x +a \right )}{b}\) | \(33\) |
parallelrisch | \(\frac {25 \sinh \left (3 b x +3 a \right )+150 \sinh \left (b x +a \right )+3 \sinh \left (5 b x +5 a \right )}{240 b}\) | \(37\) |
risch | \(\frac {{\mathrm e}^{5 b x +5 a}}{160 b}+\frac {5 \,{\mathrm e}^{3 b x +3 a}}{96 b}+\frac {5 \,{\mathrm e}^{b x +a}}{16 b}-\frac {5 \,{\mathrm e}^{-b x -a}}{16 b}-\frac {5 \,{\mathrm e}^{-3 b x -3 a}}{96 b}-\frac {{\mathrm e}^{-5 b x -5 a}}{160 b}\) | \(83\) |
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none
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61 \[ \int \cosh ^5(a+b x) \, dx=\frac {3 \, \sinh \left (b x + a\right )^{5} + 5 \, {\left (6 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right )^{3} + 15 \, {\left (\cosh \left (b x + a\right )^{4} + 5 \, \cosh \left (b x + a\right )^{2} + 10\right )} \sinh \left (b x + a\right )}{240 \, b} \]
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Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.41 \[ \int \cosh ^5(a+b x) \, dx=\begin {cases} \frac {8 \sinh ^{5}{\left (a + b x \right )}}{15 b} - \frac {4 \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} + \frac {\sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cosh ^{5}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (37) = 74\).
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.00 \[ \int \cosh ^5(a+b x) \, dx=\frac {e^{\left (5 \, b x + 5 \, a\right )}}{160 \, b} + \frac {5 \, e^{\left (3 \, b x + 3 \, a\right )}}{96 \, b} + \frac {5 \, e^{\left (b x + a\right )}}{16 \, b} - \frac {5 \, e^{\left (-b x - a\right )}}{16 \, b} - \frac {5 \, e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} - \frac {e^{\left (-5 \, b x - 5 \, a\right )}}{160 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.00 \[ \int \cosh ^5(a+b x) \, dx=\frac {e^{\left (5 \, b x + 5 \, a\right )}}{160 \, b} + \frac {5 \, e^{\left (3 \, b x + 3 \, a\right )}}{96 \, b} + \frac {5 \, e^{\left (b x + a\right )}}{16 \, b} - \frac {5 \, e^{\left (-b x - a\right )}}{16 \, b} - \frac {5 \, e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} - \frac {e^{\left (-5 \, b x - 5 \, a\right )}}{160 \, b} \]
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Time = 1.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \cosh ^5(a+b x) \, dx=\frac {\frac {{\mathrm {sinh}\left (a+b\,x\right )}^5}{5}+\frac {2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3}+\mathrm {sinh}\left (a+b\,x\right )}{b} \]
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