Integrand size = 17, antiderivative size = 39 \[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log (x)}{2}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
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Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2715, 8} \[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log (x)}{2} \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \cosh ^2(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 \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{2 n} \\ & = \frac {\log (x)}{2}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \left (a+b \log \left (c x^n\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \]
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Time = 0.90 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77
method | result | size |
parallelrisch | \(\frac {2 \ln \left (x \right ) b n +\sinh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{4 b n}\) | \(30\) |
derivativedivides | \(\frac {\frac {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}+\frac {a}{2}}{n b}\) | \(45\) |
default | \(\frac {\frac {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}+\frac {a}{2}}{n b}\) | \(45\) |
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {b n \log \left (x\right ) + \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{2 \, b n} \]
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\[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.26 \[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} + \frac {1}{2} \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.05 \[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (4 \, b c^{2 \, b} n e^{\left (2 \, a\right )} \log \left (x\right ) + c^{4 \, b} x^{2 \, b n} e^{\left (4 \, a\right )} - \frac {2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{2 \, b n}}\right )} e^{\left (-2 \, a\right )}}{8 \, b c^{2 \, b} n} \]
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Time = 1.80 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {\cosh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\ln \left (x^n\right )}{2\,n}+\frac {\mathrm {sinh}\left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4\,b\,n} \]
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