3.226 \(\int x \cosh (x) \, dx\)

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

[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 = {3296, 2638} \[ x \sinh (x)-\cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[x],x]

[Out]

-Cosh[x] + x*Sinh[x]

Rule 2638

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

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]

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 \[ x \sinh (x)-\cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[x],x]

[Out]

-Cosh[x] + x*Sinh[x]

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fricas [A]  time = 0.41, size = 9, normalized size = 1.00 \[ x \sinh \relax (x) - \cosh \relax (x) \]

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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giac [A]  time = 1.20, size = 17, normalized size = 1.89 \[ -\frac {1}{2} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {1}{2} \, {\left (x - 1\right )} e^{x} \]

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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maple [A]  time = 0.01, size = 10, normalized size = 1.11 \[ x \sinh \relax (x )-\cosh \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(x),x)

[Out]

-cosh(x)+x*sinh(x)

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maxima [B]  time = 0.43, size = 34, normalized size = 3.78 \[ \frac {1}{2} \, x^{2} \cosh \relax (x) - \frac {1}{4} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac {1}{4} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} \]

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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mupad [B]  time = 0.03, size = 9, normalized size = 1.00 \[ x\,\mathrm {sinh}\relax (x)-\mathrm {cosh}\relax (x) \]

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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sympy [A]  time = 0.19, size = 7, normalized size = 0.78 \[ x \sinh {\relax (x )} - \cosh {\relax (x )} \]

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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