Integrand size = 17, antiderivative size = 73 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \log (x)}{8}-\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2715, 8} \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {3 \log (x)}{8} \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sinh ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {3 \text {Subst}\left (\int \sinh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n} \\ & = -\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n} \\ & = \frac {3 \log (x)}{8}-\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 \left (a+b \log \left (c x^n\right )\right )-8 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]
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Time = 21.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {12 \ln \left (x \right ) b n -8 \sinh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+\sinh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )}{32 b n}\) | \(46\) |
derivativedivides | \(\frac {\left (\frac {{\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}-\frac {3 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) | \(62\) |
default | \(\frac {\left (\frac {{\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}-\frac {3 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) | \(62\) |
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Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.15 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, b n \log \left (x\right ) + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \]
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\[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {e^{\left (4 \, b \log \left (c x^{n}\right ) + 4 \, a\right )}}{64 \, b n} - \frac {e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} + \frac {e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-4 \, b \log \left (c x^{n}\right ) - 4 \, a\right )}}{64 \, b n} + \frac {3}{8} \, \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.56 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (24 \, b c^{4 \, b} n e^{\left (4 \, a\right )} \log \left (x\right ) + c^{8 \, b} x^{4 \, b n} e^{\left (8 \, a\right )} - 8 \, c^{6 \, b} x^{2 \, b n} e^{\left (6 \, a\right )} - \frac {18 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 8 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{4 \, b n}}\right )} e^{\left (-4 \, a\right )}}{64 \, b c^{4 \, b} n} \]
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Time = 1.55 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3\,\ln \left (x^n\right )}{8\,n}-\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}-\frac {\mathrm {sinh}\left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \]
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