Integrand size = 17, antiderivative size = 65 \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2713} \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n}-\frac {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sinh ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = \frac {\cosh \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {2 \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\cosh ^5\left (a+b \log \left (c x^n\right )\right )}{5 b n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05 \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {5 \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {5 \cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{48 b n}+\frac {\cosh \left (5 \left (a+b \log \left (c x^n\right )\right )\right )}{80 b n} \]
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Time = 74.46 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\left (\frac {8}{15}+\frac {{\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {4 {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{15}\right ) \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(51\) |
default | \(\frac {\left (\frac {8}{15}+\frac {{\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {4 {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{15}\right ) \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(51\) |
parallelrisch | \(\frac {128-25 \cosh \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )+150 \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )+3 \cosh \left (5 b \ln \left (c \,x^{n}\right )+5 a \right )}{240 b n}\) | \(56\) |
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (61) = 122\).
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.00 \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 15 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 25 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 15 \, {\left (2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 5 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 150 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{240 \, b n} \]
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Time = 7.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.62 \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \sinh ^{5}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \sinh ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} - \frac {4 \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{3 b n} + \frac {8 \cosh ^{5}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{15 b n} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (61) = 122\).
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.00 \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {e^{\left (5 \, b \log \left (c x^{n}\right ) + 5 \, a\right )}}{160 \, b n} - \frac {5 \, e^{\left (3 \, b \log \left (c x^{n}\right ) + 3 \, a\right )}}{96 \, b n} + \frac {5 \, e^{\left (b \log \left (c x^{n}\right ) + a\right )}}{16 \, b n} + \frac {5 \, e^{\left (-b \log \left (c x^{n}\right ) - a\right )}}{16 \, b n} - \frac {5 \, e^{\left (-3 \, b \log \left (c x^{n}\right ) - 3 \, a\right )}}{96 \, b n} + \frac {e^{\left (-5 \, b \log \left (c x^{n}\right ) - 5 \, a\right )}}{160 \, b n} \]
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Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.77 \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (3 \, c^{10 \, b} x^{5 \, b n} e^{\left (10 \, a\right )} - 25 \, c^{8 \, b} x^{3 \, b n} e^{\left (8 \, a\right )} + 150 \, c^{6 \, b} x^{b n} e^{\left (6 \, a\right )} + \frac {150 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 25 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 3}{x^{5 \, b n}}\right )} e^{\left (-5 \, a\right )}}{480 \, b c^{5 \, b} n} \]
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Time = 1.59 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int \frac {\sinh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\frac {{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^5}{5}-\frac {2\,{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3}+\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}{b\,n} \]
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