Optimal. Leaf size=19 \[ \frac {y \sinh (a y)}{a}-\frac {\cosh (a y)}{a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 2638
Rule 3296
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*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ \frac {y \sinh (a y)}{a}-\frac {\cosh (a y)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 18, normalized size = 0.95 \[ \frac {a y \sinh \left (a y\right ) - \cosh \left (a y\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 30, normalized size = 1.58 \[ \frac {{\left (a y - 1\right )} e^{\left (a y\right )}}{2 \, a^{2}} - \frac {{\left (a y + 1\right )} e^{\left (-a y\right )}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 19, normalized size = 1.00 \[ \frac {a y \sinh \left (a y \right )-\cosh \left (a y \right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 57, normalized size = 3.00 \[ \frac {1}{2} \, y^{2} \cosh \left (a y\right ) - \frac {1}{4} \, a {\left (\frac {{\left (a^{2} y^{2} - 2 \, a y + 2\right )} e^{\left (a y\right )}}{a^{3}} + \frac {{\left (a^{2} y^{2} + 2 \, a y + 2\right )} e^{\left (-a y\right )}}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 18, normalized size = 0.95 \[ -\frac {\mathrm {cosh}\left (a\,y\right )-a\,y\,\mathrm {sinh}\left (a\,y\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 20, normalized size = 1.05 \[ \begin {cases} \frac {y \sinh {\left (a y \right )}}{a} - \frac {\cosh {\left (a y \right )}}{a^{2}} & \text {for}\: a \neq 0 \\\frac {y^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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