3.48 \(\int \frac{\sinh (a+\frac{b}{x^2})}{x^3} \, dx\)

Optimal. Leaf size=15 \[ -\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b} \]

[Out]

-Cosh[a + b/x^2]/(2*b)

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Rubi [A]  time = 0.0194359, antiderivative size = 15, 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} \[ -\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x^2]/x^3,x]

[Out]

-Cosh[a + b/x^2]/(2*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^2}\right )}{x^3} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0046271, size = 15, normalized size = 1. \[ -\frac{\cosh \left (a+\frac{b}{x^2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b/x^2]/x^3,x]

[Out]

-Cosh[a + b/x^2]/(2*b)

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Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b}\cosh \left ( a+{\frac{b}{{x}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b/x^2)/x^3,x)

[Out]

-1/2*cosh(a+b/x^2)/b

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Maxima [A]  time = 1.1396, size = 18, normalized size = 1.2 \begin{align*} -\frac{\cosh \left (a + \frac{b}{x^{2}}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x^2)/x^3,x, algorithm="maxima")

[Out]

-1/2*cosh(a + b/x^2)/b

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Fricas [A]  time = 1.61495, size = 41, normalized size = 2.73 \begin{align*} -\frac{\cosh \left (\frac{a x^{2} + b}{x^{2}}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x^2)/x^3,x, algorithm="fricas")

[Out]

-1/2*cosh((a*x^2 + b)/x^2)/b

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Sympy [A]  time = 4.68851, size = 22, normalized size = 1.47 \begin{align*} \begin{cases} - \frac{\cosh{\left (a + \frac{b}{x^{2}} \right )}}{2 b} & \text{for}\: b \neq 0 \\- \frac{\sinh{\left (a \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x**2)/x**3,x)

[Out]

Piecewise((-cosh(a + b/x**2)/(2*b), Ne(b, 0)), (-sinh(a)/(2*x**2), True))

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Giac [A]  time = 1.2381, size = 34, normalized size = 2.27 \begin{align*} -\frac{e^{\left (a + \frac{b}{x^{2}}\right )} + e^{\left (-a - \frac{b}{x^{2}}\right )}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b/x^2)/x^3,x, algorithm="giac")

[Out]

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