### 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 = 0.0162827, 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.

[In]

Int[y*Cosh[a*y],y]

[Out]

-(Cosh[a*y]/a^2) + (y*Sinh[a*y])/a

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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.

[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 = 0.009, 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.

[In]

int(y*cosh(a*y),y)

[Out]

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

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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.

[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 = 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.

[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 = 0.21223, 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.

[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 [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.

[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