Integrand size = 18, antiderivative size = 95 \[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=-\frac {\left (1+e^{2 (d+e x)}\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n-b c \log (F)}{2 e},\frac {1}{2} \left (2-n+\frac {b c \log (F)}{e}\right ),-e^{2 (d+e x)}\right )}{e n-b c \log (F)} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5591, 2291} \[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=-\frac {\left (e^{2 (d+e x)}+1\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n-b c \log (F)}{2 e},\frac {1}{2} \left (-n+\frac {b c \log (F)}{e}+2\right ),-e^{2 (d+e x)}\right )}{e n-b c \log (F)} \]
[In]
[Out]
Rule 2291
Rule 5591
Rubi steps \begin{align*} \text {integral}& = \left (e^{n (d+e x)} \left (1+e^{2 (d+e x)}\right )^{-n} \cosh ^n(d+e x)\right ) \int e^{-n (d+e x)} \left (1+e^{2 (d+e x)}\right )^n F^{c (a+b x)} \, dx \\ & = -\frac {\left (1+e^{2 (d+e x)}\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,-\frac {e n-b c \log (F)}{2 e},\frac {1}{2} \left (2-n+\frac {b c \log (F)}{e}\right ),-e^{2 (d+e x)}\right )}{e n-b c \log (F)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=\frac {\left (1+e^{2 (d+e x)}\right )^{-n} F^{c (a+b x)} \cosh ^n(d+e x) \operatorname {Hypergeometric2F1}\left (-n,\frac {-e n+b c \log (F)}{2 e},1+\frac {-e n+b c \log (F)}{2 e},-e^{2 (d+e x)}\right )}{-e n+b c \log (F)} \]
[In]
[Out]
\[\int F^{c \left (b x +a \right )} \cosh \left (e x +d \right )^{n}d x\]
[In]
[Out]
\[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \cosh \left (e x + d\right )^{n} \,d x } \]
[In]
[Out]
\[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=\int F^{c \left (a + b x\right )} \cosh ^{n}{\left (d + e x \right )}\, dx \]
[In]
[Out]
\[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \cosh \left (e x + d\right )^{n} \,d x } \]
[In]
[Out]
\[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=\int { F^{{\left (b x + a\right )} c} \cosh \left (e x + d\right )^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int F^{c (a+b x)} \cosh ^n(d+e x) \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\mathrm {cosh}\left (d+e\,x\right )}^n \,d x \]
[In]
[Out]