Optimal. Leaf size=15 \[ a x+\frac{b \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.0084691, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2637} \[ a x+\frac{b \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x)) \, dx &=a x+b \int \cosh (c+d x) \, dx\\ &=a x+\frac{b \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0082455, size = 26, normalized size = 1.73 \[ a x+\frac{b \sinh (c) \cosh (d x)}{d}+\frac{b \cosh (c) \sinh (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{b\sinh \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0172, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{b \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09114, size = 39, normalized size = 2.6 \begin{align*} \frac{a d x + b \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.142993, size = 17, normalized size = 1.13 \begin{align*} a x + b \left (\begin{cases} \frac{\sinh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \cosh{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17885, size = 43, normalized size = 2.87 \begin{align*} a x + \frac{1}{2} \, b{\left (\frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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