3.34 \(\int y \sinh (y) \, dy\)

Optimal. Leaf size=9 \[ y \cosh (y)-\sinh (y) \]

[Out]

y*Cosh[y] - Sinh[y]

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Rubi [A]  time = 0.0115834, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3296, 2637} \[ y \cosh (y)-\sinh (y) \]

Antiderivative was successfully verified.

[In]

Int[y*Sinh[y],y]

[Out]

y*Cosh[y] - Sinh[y]

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int y \sinh (y) \, dy &=y \cosh (y)-\int \cosh (y) \, dy\\ &=y \cosh (y)-\sinh (y)\\ \end{align*}

Mathematica [A]  time = 0.0031957, size = 9, normalized size = 1. \[ y \cosh (y)-\sinh (y) \]

Antiderivative was successfully verified.

[In]

Integrate[y*Sinh[y],y]

[Out]

y*Cosh[y] - Sinh[y]

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Maple [A]  time = 0.001, size = 10, normalized size = 1.1 \begin{align*} y\cosh \left ( y \right ) -\sinh \left ( y \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(y*sinh(y),y)

[Out]

y*cosh(y)-sinh(y)

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Maxima [B]  time = 0.948028, size = 46, normalized size = 5.11 \begin{align*} \frac{1}{2} \, y^{2} \sinh \left (y\right ) + \frac{1}{4} \,{\left (y^{2} + 2 \, y + 2\right )} e^{\left (-y\right )} - \frac{1}{4} \,{\left (y^{2} - 2 \, y + 2\right )} e^{y} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(y*sinh(y),y, algorithm="maxima")

[Out]

1/2*y^2*sinh(y) + 1/4*(y^2 + 2*y + 2)*e^(-y) - 1/4*(y^2 - 2*y + 2)*e^y

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Fricas [A]  time = 1.78134, size = 28, normalized size = 3.11 \begin{align*} y \cosh \left (y\right ) - \sinh \left (y\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(y*sinh(y),y, algorithm="fricas")

[Out]

y*cosh(y) - sinh(y)

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Sympy [A]  time = 0.171068, size = 7, normalized size = 0.78 \begin{align*} y \cosh{\left (y \right )} - \sinh{\left (y \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(y*sinh(y),y)

[Out]

y*cosh(y) - sinh(y)

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Giac [A]  time = 1.04867, size = 23, normalized size = 2.56 \begin{align*} \frac{1}{2} \,{\left (y + 1\right )} e^{\left (-y\right )} + \frac{1}{2} \,{\left (y - 1\right )} e^{y} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(y*sinh(y),y, algorithm="giac")

[Out]

1/2*(y + 1)*e^(-y) + 1/2*(y - 1)*e^y