Optimal. Leaf size=15 \[ a x+\frac {b \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.39, size = 17, normalized size = 1.13 \[ \frac {a d x + b \sinh \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 32, normalized size = 2.13 \[ a x + \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 16, normalized size = 1.07 \[ a x +\frac {b \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.29, size = 15, normalized size = 1.00 \[ a x + \frac {b \sinh \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 15, normalized size = 1.00 \[ a\,x+\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 17, normalized size = 1.13 \[ a x + b \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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