3.1.34 \(\int \frac {\sinh (a+\frac {b}{x})}{x^3} \, dx\) [34]

Optimal. Leaf size=29 \[ -\frac {\cosh \left (a+\frac {b}{x}\right )}{b x}+\frac {\sinh \left (a+\frac {b}{x}\right )}{b^2} \]

[Out]

-cosh(a+b/x)/b/x+sinh(a+b/x)/b^2

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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5428, 3377, 2717} \begin {gather*} \frac {\sinh \left (a+\frac {b}{x}\right )}{b^2}-\frac {\cosh \left (a+\frac {b}{x}\right )}{b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x]/x^3,x]

[Out]

-(Cosh[a + b/x]/(b*x)) + Sinh[a + b/x]/b^2

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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

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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {-b \cosh \left (a+\frac {b}{x}\right )+x \sinh \left (a+\frac {b}{x}\right )}{b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b/x]/x^3,x]

[Out]

(-(b*Cosh[a + b/x]) + x*Sinh[a + b/x])/(b^2*x)

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Maple [A]
time = 0.72, size = 44, normalized size = 1.52

method	result	size
derivativedivides	\(-\frac {\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )-a \cosh \left (a +\frac {b}{x}\right )}{b^{2}}\)	\(44\)
default	\(-\frac {\left (a +\frac {b}{x}\right ) \cosh \left (a +\frac {b}{x}\right )-\sinh \left (a +\frac {b}{x}\right )-a \cosh \left (a +\frac {b}{x}\right )}{b^{2}}\)	\(44\)
risch	\(-\frac {\left (-x +b \right ) {\mathrm e}^{\frac {a x +b}{x}}}{2 b^{2} x}-\frac {\left (x +b \right ) {\mathrm e}^{-\frac {a x +b}{x}}}{2 b^{2} x}\)	\(47\)
meijerg	\(-\frac {\cosh \left (a \right ) \left (\frac {\cosh \left (\frac {b}{x}\right ) b}{x}-\sinh \left (\frac {b}{x}\right )\right )}{b^{2}}+\frac {2 \sqrt {\pi }\, \sinh \left (a \right ) \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }}-\frac {b \sinh \left (\frac {b}{x}\right )}{2 \sqrt {\pi }\, x}\right )}{b^{2}}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b/x)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.30, size = 48, normalized size = 1.66 \begin {gather*} -\frac {1}{4} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (3, \frac {b}{x}\right )}{b^{3}} - \frac {e^{a} \Gamma \left (3, -\frac {b}{x}\right )}{b^{3}}\right )} - \frac {\sinh \left (a + \frac {b}{x}\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*(e^(-a)*gamma(3, b/x)/b^3 - e^a*gamma(3, -b/x)/b^3) - 1/2*sinh(a + b/x)/x^2

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Fricas [A]
time = 0.41, size = 34, normalized size = 1.17 \begin {gather*} -\frac {b \cosh \left (\frac {a x + b}{x}\right ) - x \sinh \left (\frac {a x + b}{x}\right )}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.47, size = 29, normalized size = 1.00 \begin {gather*} \begin {cases} - \frac {\cosh {\left (a + \frac {b}{x} \right )}}{b x} + \frac {\sinh {\left (a + \frac {b}{x} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- \frac {\sinh {\left (a \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (29) = 58\).
time = 0.42, size = 95, normalized size = 3.28 \begin {gather*} \frac {a e^{\left (\frac {a x + b}{x}\right )} + a e^{\left (-\frac {a x + b}{x}\right )} - \frac {{\left (a x + b\right )} e^{\left (\frac {a x + b}{x}\right )}}{x} - \frac {{\left (a x + b\right )} e^{\left (-\frac {a x + b}{x}\right )}}{x} + e^{\left (\frac {a x + b}{x}\right )} - e^{\left (-\frac {a x + b}{x}\right )}}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.38, size = 29, normalized size = 1.00 \begin {gather*} \frac {\mathrm {sinh}\left (a+\frac {b}{x}\right )}{b^2}-\frac {\mathrm {cosh}\left (a+\frac {b}{x}\right )}{b\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b/x)/x^3,x)

[Out]

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

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