3.33 \(\int \frac{\sinh (a+\frac{b}{x})}{x^2} \, dx\)

Optimal. Leaf size=13 \[ -\frac{\cosh \left (a+\frac{b}{x}\right )}{b} \]

[Out]

-(Cosh[a + b/x]/b)

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Rubi [A]  time = 0.0169405, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5320, 2638} \[ -\frac{\cosh \left (a+\frac{b}{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x]/x^2,x]

[Out]

-(Cosh[a + b/x]/b)

Rule 5320

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 2638

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

Rubi steps

\begin{align*} \int \frac{\sinh \left (a+\frac{b}{x}\right )}{x^2} \, dx &=-\operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\cosh \left (a+\frac{b}{x}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0045134, size = 13, normalized size = 1. \[ -\frac{\cosh \left (a+\frac{b}{x}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b/x]/x^2,x]

[Out]

-(Cosh[a + b/x]/b)

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Maple [A]  time = 0.004, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{b}\cosh \left ( a+{\frac{b}{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b/x)/x^2,x)

[Out]

-cosh(a+b/x)/b

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Maxima [A]  time = 1.05249, size = 18, normalized size = 1.38 \begin{align*} -\frac{\cosh \left (a + \frac{b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x)/x^2,x, algorithm="maxima")

[Out]

-cosh(a + b/x)/b

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Fricas [A]  time = 1.62389, size = 30, normalized size = 2.31 \begin{align*} -\frac{\cosh \left (\frac{a x + b}{x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x)/x^2,x, algorithm="fricas")

[Out]

-cosh((a*x + b)/x)/b

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Sympy [A]  time = 1.56414, size = 15, normalized size = 1.15 \begin{align*} \begin{cases} - \frac{\cosh{\left (a + \frac{b}{x} \right )}}{b} & \text{for}\: b \neq 0 \\- \frac{\sinh{\left (a \right )}}{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x)/x**2,x)

[Out]

Piecewise((-cosh(a + b/x)/b, Ne(b, 0)), (-sinh(a)/x, True))

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Giac [A]  time = 1.25665, size = 34, normalized size = 2.62 \begin{align*} -\frac{e^{\left (a + \frac{b}{x}\right )} + e^{\left (-a - \frac{b}{x}\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x)/x^2,x, algorithm="giac")

[Out]

-1/2*(e^(a + b/x) + e^(-a - b/x))/b