Integrand size = 20, antiderivative size = 95 \[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=-\frac {x^{-n} (e x)^n \left (b \cosh \left (c+d x^n\right )\right )^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\cosh ^2\left (c+d x^n\right )\right ) \sinh \left (c+d x^n\right )}{b d e n (1+p) \sqrt {-\sinh ^2\left (c+d x^n\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5431, 5429, 2722} \[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=-\frac {x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (b \cosh \left (c+d x^n\right )\right )^{p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {p+1}{2},\frac {p+3}{2},\cosh ^2\left (d x^n+c\right )\right )}{b d e n (p+1) \sqrt {-\sinh ^2\left (c+d x^n\right )}} \]
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Rule 2722
Rule 5429
Rule 5431
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx}{e} \\ & = \frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int (b \cosh (c+d x))^p \, dx,x,x^n\right )}{e n} \\ & = -\frac {x^{-n} (e x)^n \left (b \cosh \left (c+d x^n\right )\right )^{1+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\cosh ^2\left (c+d x^n\right )\right ) \sinh \left (c+d x^n\right )}{b d e n (1+p) \sqrt {-\sinh ^2\left (c+d x^n\right )}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=\frac {x^{1-n} (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \coth \left (c+d x^n\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\cosh ^2\left (c+d x^n\right )\right ) \sqrt {-\sinh ^2\left (c+d x^n\right )}}{d n (1+p)} \]
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\[\int \left (e x \right )^{-1+n} {\left (b \cosh \left (c +d \,x^{n}\right )\right )}^{p}d x\]
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\[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p} \,d x } \]
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\[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=\int \left (b \cosh {\left (c + d x^{n} \right )}\right )^{p} \left (e x\right )^{n - 1}\, dx \]
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\[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p} \,d x } \]
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\[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} \left (b \cosh \left (d x^{n} + c\right )\right )^{p} \,d x } \]
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Timed out. \[ \int (e x)^{-1+n} \left (b \cosh \left (c+d x^n\right )\right )^p \, dx=\int {\left (e\,x\right )}^{n-1}\,{\left (b\,\mathrm {cosh}\left (c+d\,x^n\right )\right )}^p \,d x \]
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