Optimal. Leaf size=75 \[ \frac{e \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac{b c \log (F) \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \]
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Rubi [A] time = 0.0182983, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {5475} \[ \frac{e \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac{b c \log (F) \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \]
Antiderivative was successfully verified.
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Rule 5475
Rubi steps
\begin{align*} \int F^{c (a+b x)} \cosh (d+e x) \, dx &=-\frac{b c F^{c (a+b x)} \cosh (d+e x) \log (F)}{e^2-b^2 c^2 \log ^2(F)}+\frac{e F^{c (a+b x)} \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}\\ \end{align*}
Mathematica [A] time = 0.103601, size = 50, normalized size = 0.67 \[ \frac{F^{c (a+b x)} (e \sinh (d+e x)-b c \log (F) \cosh (d+e x))}{(e-b c \log (F)) (b c \log (F)+e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 74, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( F \right ) bc{{\rm e}^{2\,ex+2\,d}}+bc\ln \left ( F \right ) -e{{\rm e}^{2\,ex+2\,d}}+e \right ){{\rm e}^{-ex-d}}{F}^{c \left ( bx+a \right ) }}{ \left ( 2\,bc\ln \left ( F \right ) -2\,e \right ) \left ( e+bc\ln \left ( F \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04999, size = 85, normalized size = 1.13 \begin{align*} \frac{F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{2 \,{\left (b c \log \left (F\right ) + e\right )}} + \frac{F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{2 \,{\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.51874, size = 643, normalized size = 8.57 \begin{align*} -\frac{{\left (e \cosh \left (e x + d\right )^{2} -{\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} -{\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \left (F\right ) - 2 \,{\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) +{\left (e \cosh \left (e x + d\right )^{2} -{\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} -{\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \left (F\right ) - 2 \,{\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{2 \,{\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - e^{2} \cosh \left (e x + d\right ) +{\left (b^{2} c^{2} \log \left (F\right )^{2} - e^{2}\right )} \sinh \left (e x + d\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.0146, size = 316, normalized size = 4.21 \begin{align*} \begin{cases} - \frac{\left (-1\right )^{a c} \left (-1\right )^{- \frac{i e x}{\pi }} x \sinh{\left (d + e x \right )}}{2} + \frac{\left (-1\right )^{a c} \left (-1\right )^{- \frac{i e x}{\pi }} x \cosh{\left (d + e x \right )}}{2} + \frac{\left (-1\right )^{a c} \left (-1\right )^{- \frac{i e x}{\pi }} \sinh{\left (d + e x \right )}}{2 e} & \text{for}\: F = -1 \wedge b = - \frac{i e}{\pi c} \\x \cosh{\left (d \right )} & \text{for}\: F = 1 \wedge e = 0 \\\tilde{\infty } e \left (e^{- \frac{e}{b c}}\right )^{a c} \left (e^{- \frac{e}{b c}}\right )^{b c x} \sinh{\left (d + e x \right )} + \tilde{\infty } e \left (e^{- \frac{e}{b c}}\right )^{a c} \left (e^{- \frac{e}{b c}}\right )^{b c x} \cosh{\left (d + e x \right )} & \text{for}\: F = e^{- \frac{e}{b c}} \\\tilde{\infty } e \left (e^{\frac{e}{b c}}\right )^{a c} \left (e^{\frac{e}{b c}}\right )^{b c x} \sinh{\left (d + e x \right )} + \tilde{\infty } e \left (e^{\frac{e}{b c}}\right )^{a c} \left (e^{\frac{e}{b c}}\right )^{b c x} \cosh{\left (d + e x \right )} & \text{for}\: F = e^{\frac{e}{b c}} \\\frac{F^{a c} F^{b c x} b c \log{\left (F \right )} \cosh{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} - e^{2}} - \frac{F^{a c} F^{b c x} e \sinh{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} - e^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.3714, size = 825, normalized size = 11. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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