3.225 \(\int x \sinh (x) \, dx\)

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

[Out]

x*Cosh[x] - Sinh[x]

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Rubi [A]  time = 0.0109491, 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} \[ x \cosh (x)-\sinh (x) \]

Antiderivative was successfully verified.

[In]

Int[x*Sinh[x],x]

[Out]

x*Cosh[x] - Sinh[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]

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

Mathematica [A]  time = 0.0023825, size = 9, normalized size = 1. \[ x \cosh (x)-\sinh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sinh[x],x]

[Out]

x*Cosh[x] - Sinh[x]

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Maple [A]  time = 0.005, size = 10, normalized size = 1.1 \begin{align*} x\cosh \left ( x \right ) -\sinh \left ( x \right ) \end{align*}

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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Maxima [B]  time = 0.951741, size = 46, normalized size = 5.11 \begin{align*} \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{align*}

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 = 1.56237, size = 28, normalized size = 3.11 \begin{align*} x \cosh \left (x\right ) - \sinh \left (x\right ) \end{align*}

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 = 0.172455, size = 7, normalized size = 0.78 \begin{align*} x \cosh{\left (x \right )} - \sinh{\left (x \right )} \end{align*}

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 [A]  time = 1.08537, size = 23, normalized size = 2.56 \begin{align*} \frac{1}{2} \,{\left (x + 1\right )} e^{\left (-x\right )} + \frac{1}{2} \,{\left (x - 1\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x + 1)*e^(-x) + 1/2*(x - 1)*e^x