3.35 \(\int y \cosh (a y) \, dy\)

Optimal. Leaf size=19 \[ \frac{y \sinh (a y)}{a}-\frac{\cosh (a y)}{a^2} \]

[Out]

-(Cosh[a*y]/a^2) + (y*Sinh[a*y])/a

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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]

[Out]

-(Cosh[a*y]/a^2) + (y*Sinh[a*y])/a

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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)

[Out]

y*sinh(a*y)/a - Integral(sinh(a*y), y)/a

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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]

[Out]

-(Cosh[a*y]/a^2) + (y*Sinh[a*y])/a

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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)

[Out]

1/a^2*(y*a*sinh(a*y)-cosh(a*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")

[Out]

1/2*y^2*cosh(a*y) - 1/4*a*((a^2*y^2 - 2*a*y + 2)*e^(a*y)/a^3 + (a^2*y^2 + 2*a*y
+ 2)*e^(-a*y)/a^3)

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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")

[Out]

(a*y*sinh(a*y) - cosh(a*y))/a^2

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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)

[Out]

Piecewise((y*sinh(a*y)/a - cosh(a*y)/a**2, Ne(a, 0)), (y**2/2, True))

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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]

1/2*(a*y - 1)*e^(a*y)/a^2 - 1/2*(a*y + 1)*e^(-a*y)/a^2