3.1.33 \(\int \frac {\sinh (a+\frac {b}{x})}{x^2} \, dx\) [33]

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 = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = {5428, 2718} \begin {gather*} -\frac {\cosh \left (a+\frac {b}{x}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x]/x^2,x]

[Out]

-(Cosh[a + b/x]/b)

Rule 2718

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

Rule 5428

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]))

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\cosh \left (a+\frac {b}{x}\right )}{b} \end {gather*}

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 = 0.24, size = 14, normalized size = 1.08

method	result	size
derivativedivides	\(-\frac {\cosh \left (a +\frac {b}{x}\right )}{b}\)	\(14\)
default	\(-\frac {\cosh \left (a +\frac {b}{x}\right )}{b}\)	\(14\)
risch	\(-\frac {{\mathrm e}^{\frac {a x +b}{x}}}{2 b}-\frac {{\mathrm e}^{-\frac {a x +b}{x}}}{2 b}\)	\(33\)
meijerg	\(\frac {\sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (\frac {b}{x}\right )}{\sqrt {\pi }}\right )}{b}-\frac {\sinh \left (a \right ) \sinh \left (\frac {b}{x}\right )}{b}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cosh(a+b/x)/b

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Maxima [A]
time = 0.26, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\cosh \left (a + \frac {b}{x}\right )}{b} \end {gather*}

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 = 0.32, size = 15, normalized size = 1.15 \begin {gather*} -\frac {\cosh \left (\frac {a x + b}{x}\right )}{b} \end {gather*}

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 = 0.32, size = 15, normalized size = 1.15 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).
time = 0.45, size = 27, normalized size = 2.08 \begin {gather*} -\frac {e^{\left (\frac {a x + b}{x}\right )} + e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b} \end {gather*}

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*x + b)/x) + e^(-(a*x + b)/x))/b

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Mupad [B]
time = 0.37, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {cosh}\left (a+\frac {b}{x}\right )}{b} \end {gather*}

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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