Optimal. Leaf size=19 \[ \frac{y \sinh (a y)}{a}-\frac{\cosh (a y)}{a^2} \]
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Rubi [A] time = 0.0162827, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3296, 2638} \[ \frac{y \sinh (a y)}{a}-\frac{\cosh (a y)}{a^2} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int y \cosh (a y) \, dy &=\frac{y \sinh (a y)}{a}-\frac{\int \sinh (a y) \, dy}{a}\\ &=-\frac{\cosh (a y)}{a^2}+\frac{y \sinh (a y)}{a}\\ \end{align*}
Mathematica [A] time = 0.0129749, size = 19, normalized size = 1. \[ \frac{y \sinh (a y)}{a}-\frac{\cosh (a y)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 19, normalized size = 1. \begin{align*}{\frac{ya\sinh \left ( ay \right ) -\cosh \left ( ay \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.950136, size = 77, normalized size = 4.05 \begin{align*} \frac{1}{2} \, y^{2} \cosh \left (a y\right ) - \frac{1}{4} \, a{\left (\frac{{\left (a^{2} y^{2} - 2 \, a y + 2\right )} e^{\left (a y\right )}}{a^{3}} + \frac{{\left (a^{2} y^{2} + 2 \, a y + 2\right )} e^{\left (-a y\right )}}{a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77083, size = 45, normalized size = 2.37 \begin{align*} \frac{a y \sinh \left (a y\right ) - \cosh \left (a y\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.21223, size = 20, normalized size = 1.05 \begin{align*} \begin{cases} \frac{y \sinh{\left (a y \right )}}{a} - \frac{\cosh{\left (a y \right )}}{a^{2}} & \text{for}\: a \neq 0 \\\frac{y^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05336, size = 41, normalized size = 2.16 \begin{align*} \frac{{\left (a y - 1\right )} e^{\left (a y\right )}}{2 \, a^{2}} - \frac{{\left (a y + 1\right )} e^{\left (-a y\right )}}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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