Optimal. Leaf size=32 \[ -\frac{F^{a+b x} \left (e^{-c-d x}\right )^n}{d n-b \log (F)} \]
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Rubi [A] time = 0.0946486, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {5648, 2281, 2287, 2194} \[ -\frac{F^{a+b x} \left (e^{-c-d x}\right )^n}{d n-b \log (F)} \]
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 = 0.0720742, size = 37, normalized size = 1.16 \[ -\frac{F^{a+b x} (\cosh (c+d x)-\sinh (c+d x))^n}{d n-b \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 37, normalized size = 1.2 \begin{align*}{\frac{{F}^{bx+a} \left ( \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) ^{n}}{b\ln \left ( F \right ) -dn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15712, size = 49, normalized size = 1.53 \begin{align*} -\frac{F^{a} e^{\left (-d n x + b x \log \left (F\right )\right )}}{d n e^{\left (c n\right )} - b e^{\left (c n\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11387, size = 201, normalized size = 6.28 \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 = 8.52737, size = 92, normalized size = 2.88 \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 = 1.16676, size = 381, normalized size = 11.91 \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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