3.3.26 \(\int x \cosh (x) \, dx\) [226]

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

[Out]

-cosh(x)+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, 2718} \begin {gather*} x \sinh (x)-\cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Cosh[x],x]

[Out]

-Cosh[x] + x*Sinh[x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 \cosh (x) \, dx &=x \sinh (x)-\int \sinh (x) \, dx\\ &=-\cosh (x)+x \sinh (x)\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[x],x]

[Out]

-Cosh[x] + x*Sinh[x]

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

method	result	size
default	\(-\cosh \left (x \right )+x \sinh \left (x \right )\)	\(10\)
risch	\(\left (-\frac {1}{2}+\frac {x}{2}\right ) {\mathrm e}^{x}+\left (-\frac {1}{2}-\frac {x}{2}\right ) {\mathrm e}^{-x}\)	\(20\)
meijerg	\(-2 \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (x \right )}{2 \sqrt {\pi }}-\frac {x \sinh \left (x \right )}{2 \sqrt {\pi }}\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cosh(x)+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 = 1.31, size = 34, normalized size = 3.78 \begin {gather*} \frac {1}{2} \, x^{2} \cosh \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*cosh(x),x, algorithm="maxima")

[Out]

1/2*x^2*cosh(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 = 0.63, size = 9, normalized size = 1.00 \begin {gather*} x \sinh \left (x\right ) - \cosh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sinh(x) - cosh(x)

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Sympy [A]
time = 0.06, size = 7, normalized size = 0.78 \begin {gather*} x \sinh {\left (x \right )} - \cosh {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(x),x)

[Out]

x*sinh(x) - cosh(x)

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Giac [A]
time = 0.54, size = 17, normalized size = 1.89 \begin {gather*} -\frac {1}{2} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {1}{2} \, {\left (x - 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(x),x, algorithm="giac")

[Out]

-1/2*(x + 1)*e^(-x) + 1/2*(x - 1)*e^x

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Mupad [B]
time = 0.03, size = 9, normalized size = 1.00 \begin {gather*} x\,\mathrm {sinh}\left (x\right )-\mathrm {cosh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(x),x)

[Out]

x*sinh(x) - cosh(x)

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