3.3.25 \(\int x \sinh (x) \, dx\) [225]

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

[Out]

x*cosh(x)-sinh(x)

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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*} x \cosh (x)-\sinh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sinh[x],x]

[Out]

x*Cosh[x] - Sinh[x]

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 x \sinh (x) \, dx &=x \cosh (x)-\int \cosh (x) \, dx\\ &=x \cosh (x)-\sinh (x)\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[x*Sinh[x],x]

[Out]

x*Cosh[x] - Sinh[x]

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Mathics [A]
time = 1.78, size = 9, normalized size = 1.00 \begin {gather*} x \text {Cosh}\left [x\right ]-\text {Sinh}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*Sinh[x],x]')

[Out]

x Cosh[x] - Sinh[x]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*cosh(x)-sinh(x)

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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 = 0.26, size = 34, normalized size = 3.78 \begin {gather*} \frac {1}{2} \, x^{2} \sinh \left (x\right ) + \frac {1}{4} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac {1}{4} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.30, size = 9, normalized size = 1.00 \begin {gather*} x \cosh \left (x\right ) - \sinh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*cosh(x) - sinh(x)

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Sympy [A]
time = 0.08, size = 7, normalized size = 0.78 \begin {gather*} x \cosh {\left (x \right )} - \sinh {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(x),x)

[Out]

x*cosh(x) - sinh(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (9) = 18\).
time = 0.00, size = 17, normalized size = 1.89 \begin {gather*} \frac {\left (x-1\right ) \mathrm {e}^{x}+\left (x+1\right ) \mathrm {e}^{-x}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(x),x)

[Out]

1/2*e^x*(x - 1) + 1/2*(x + 1)/e^x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(x),x)

[Out]

x*cosh(x) - sinh(x)

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