Optimal. Leaf size=12 \[ a x+a \sinh (x)+c \cosh (x) \]
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Rubi [A] time = 0.0091605, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2637, 2638} \[ a x+a \sinh (x)+c \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (a+a \cosh (x)+c \sinh (x)) \, dx &=a x+a \int \cosh (x) \, dx+c \int \sinh (x) \, dx\\ &=a x+c \cosh (x)+a \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0019364, size = 12, normalized size = 1. \[ a x+a \sinh (x)+c \cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 13, normalized size = 1.1 \begin{align*} ax+c\cosh \left ( x \right ) +a\sinh \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998106, size = 16, normalized size = 1.33 \begin{align*} a x + c \cosh \left (x\right ) + a \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32872, size = 39, normalized size = 3.25 \begin{align*} a x + c \cosh \left (x\right ) + a \sinh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.27556, size = 12, normalized size = 1. \begin{align*} a x + a \sinh{\left (x \right )} + c \cosh{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13995, size = 35, normalized size = 2.92 \begin{align*} a x + \frac{1}{2} \, c{\left (e^{\left (-x\right )} + e^{x}\right )} - \frac{1}{2} \, a{\left (e^{\left (-x\right )} - e^{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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