Optimal. Leaf size=19 \[ \frac{y \sinh (a y)}{a}-\frac{\cosh (a y)}{a^2} \]
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Rubi [A] time = 0.0289063, 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 \[ \frac{y \sinh (a y)}{a}-\frac{\cosh (a y)}{a^2} \]
Antiderivative was successfully verified.
[In] Int[y*Cosh[a*y],y]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{y \sinh{\left (a y \right )}}{a} - \frac{\int \sinh{\left (a y \right )}\, dy}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(y*cosh(a*y),y)
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Mathematica [A] time = 0.00822932, size = 19, normalized size = 1. \[ \frac{y \sinh (a y)}{a}-\frac{\cosh (a y)}{a^2} \]
Antiderivative was successfully verified.
[In] Integrate[y*Cosh[a*y],y]
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Maple [A] time = 0.007, size = 19, normalized size = 1. \[{\frac{ya\sinh \left ( ay \right ) -\cosh \left ( ay \right ) }{{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(y*cosh(a*y),y)
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Maxima [A] time = 1.39774, size = 77, normalized size = 4.05 \[ \frac{1}{2} \, y^{2} \cosh \left (a y\right ) - \frac{1}{4} \, a{\left (\frac{{\left (a^{2} y^{2} - 2 \, a y + 2\right )} e^{\left (a y\right )}}{a^{3}} + \frac{{\left (a^{2} y^{2} + 2 \, a y + 2\right )} e^{\left (-a y\right )}}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(y*cosh(a*y),y, algorithm="maxima")
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Fricas [A] time = 0.207366, size = 24, normalized size = 1.26 \[ \frac{a y \sinh \left (a y\right ) - \cosh \left (a y\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(y*cosh(a*y),y, algorithm="fricas")
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Sympy [A] time = 0.253345, 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.
[In] integrate(y*cosh(a*y),y)
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GIAC/XCAS [A] time = 0.197812, size = 41, normalized size = 2.16 \[ \frac{{\left (a y - 1\right )} e^{\left (a y\right )}}{2 \, a^{2}} - \frac{{\left (a y + 1\right )} e^{\left (-a y\right )}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(y*cosh(a*y),y, algorithm="giac")
[Out]