3.1.34 \(\int y \sinh (y) \, dy\) [34]

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

[Out]

y*cosh(y)-sinh(y)

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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3377, 2717} \begin {gather*} y \cosh (y)-\sinh (y) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[y*Sinh[y],y]

[Out]

y*Cosh[y] - Sinh[y]

Rule 2717

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

Rule 3377

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} y \cosh (y)-\sinh (y) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[y*Sinh[y],y]

[Out]

y*Cosh[y] - Sinh[y]

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Maple [A]
time = 0.02, size = 10, normalized size = 1.11

method	result	size
default	\(y \cosh \left (y \right )-\sinh \left (y \right )\)	\(10\)
meijerg	\(y \cosh \left (y \right )-\sinh \left (y \right )\)	\(10\)
risch	\(\left (-\frac {1}{2}+\frac {y}{2}\right ) {\mathrm e}^{y}+\left (\frac {1}{2}+\frac {y}{2}\right ) {\mathrm e}^{-y}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(y*sinh(y),y,method=_RETURNVERBOSE)

[Out]

y*cosh(y)-sinh(y)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (9) = 18\).
time = 1.56, size = 34, normalized size = 3.78 \begin {gather*} \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 {gather*}

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 = 0.71, size = 9, normalized size = 1.00 \begin {gather*} y \cosh \left (y\right ) - \sinh \left (y\right ) \end {gather*}

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 = 0.06, size = 7, normalized size = 0.78 \begin {gather*} y \cosh {\left (y \right )} - \sinh {\left (y \right )} \end {gather*}

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 = 0.89, size = 17, normalized size = 1.89 \begin {gather*} \frac {1}{2} \, {\left (y + 1\right )} e^{\left (-y\right )} + \frac {1}{2} \, {\left (y - 1\right )} e^{y} \end {gather*}

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

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Mupad [B]
time = 0.02, size = 9, normalized size = 1.00 \begin {gather*} y\,\mathrm {cosh}\left (y\right )-\mathrm {sinh}\left (y\right ) \end {gather*}

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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