Optimal. Leaf size=27 \[ \frac{F^{a+b x} \left (e^{c+d x}\right )^n}{b \log (F)+d n} \]
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Rubi [A] time = 0.0910117, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5648, 2281, 2287, 2194} \[ \frac{F^{a+b x} \left (e^{c+d x}\right )^n}{b \log (F)+d n} \]
Antiderivative was successfully verified.
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Rule 5648
Rule 2281
Rule 2287
Rule 2194
Rubi steps
\begin{align*} \int F^{a+b x} (\cosh (c+d x)+\sinh (c+d x))^n \, dx &=\int \left (e^{c+d x}\right )^n F^{a+b x} \, dx\\ &=\left (e^{-n (c+d x)} \left (e^{c+d x}\right )^n\right ) \int e^{n (c+d x)} F^{a+b x} \, dx\\ &=\left (e^{-n (c+d x)} \left (e^{c+d x}\right )^n\right ) \int e^{c n+a \log (F)+x (d n+b \log (F))} \, dx\\ &=\frac{\left (e^{c+d x}\right )^n F^{a+b x}}{d n+b \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0861154, size = 33, normalized size = 1.22 \[ \frac{F^{a+b x} (\sinh (c+d x)+\cosh (c+d x))^n}{b \log (F)+d n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 34, normalized size = 1.3 \begin{align*}{\frac{{F}^{bx+a} \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) ^{n}}{dn+b\ln \left ( F \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05323, size = 38, normalized size = 1.41 \begin{align*} \frac{F^{a} e^{\left (d n x + b x \log \left (F\right ) + c n\right )}}{d n + b \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07509, size = 200, normalized size = 7.41 \begin{align*} \frac{{\left (\cosh \left (d n x + c n\right ) + \sinh \left (d n x + c n\right )\right )} \cosh \left ({\left (b x + a\right )} \log \left (F\right )\right ) +{\left (\cosh \left (d n x + c n\right ) + \sinh \left (d n x + c n\right )\right )} \sinh \left ({\left (b x + a\right )} \log \left (F\right )\right )}{d n + b \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.38276, size = 94, normalized size = 3.48 \begin{align*} \begin{cases} \frac{F^{a} F^{b x} \left (\sinh{\left (c + d x \right )} + \cosh{\left (c + d x \right )}\right )^{n}}{b \log{\left (F \right )} + d n} & \text{for}\: b \neq - \frac{d n}{\log{\left (F \right )}} \\F^{a} x \left (\sinh{\left (c + d x \right )} + \cosh{\left (c + d x \right )}\right )^{n} e^{- d n x} + \frac{F^{a} \left (\sinh{\left (c + d x \right )} + \cosh{\left (c + d x \right )}\right )^{n} e^{- d n x}}{d n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.2019, size = 370, normalized size = 13.7 \begin{align*} 2 \,{\left (\frac{2 \,{\left (d n + b \log \left ({\left | F \right |}\right )\right )} \cos \left (-\frac{1}{2} \, \pi b x \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi b x - \frac{1}{2} \, \pi a \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi a\right )}{{\left (\pi b \mathrm{sgn}\left (F\right ) - \pi b\right )}^{2} + 4 \,{\left (d n + b \log \left ({\left | F \right |}\right )\right )}^{2}} - \frac{{\left (\pi b \mathrm{sgn}\left (F\right ) - \pi b\right )} \sin \left (-\frac{1}{2} \, \pi b x \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi b x - \frac{1}{2} \, \pi a \mathrm{sgn}\left (F\right ) + \frac{1}{2} \, \pi a\right )}{{\left (\pi b \mathrm{sgn}\left (F\right ) - \pi b\right )}^{2} + 4 \,{\left (d n + b \log \left ({\left | F \right |}\right )\right )}^{2}}\right )} e^{\left (c n +{\left (d n + b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} - \frac{1}{2} i \,{\left (-\frac{2 i \, e^{\left (\frac{1}{2} i \, \pi b x \mathrm{sgn}\left (F\right ) - \frac{1}{2} i \, \pi b x + \frac{1}{2} i \, \pi a \mathrm{sgn}\left (F\right ) - \frac{1}{2} i \, \pi a\right )}}{i \, \pi b \mathrm{sgn}\left (F\right ) - i \, \pi b + 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )} + \frac{2 i \, e^{\left (-\frac{1}{2} i \, \pi b x \mathrm{sgn}\left (F\right ) + \frac{1}{2} i \, \pi b x - \frac{1}{2} i \, \pi a \mathrm{sgn}\left (F\right ) + \frac{1}{2} i \, \pi a\right )}}{-i \, \pi b \mathrm{sgn}\left (F\right ) + i \, \pi b + 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )}\right )} e^{\left (c n +{\left (d n + b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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