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

Optimal. Leaf size=57 \[ \frac {\sqrt {\pi } e^{-a} \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^a \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}} \]

[Out]

1/4*erf(b^(1/2)/x)*Pi^(1/2)/exp(a)/b^(1/2)-1/4*exp(a)*erfi(b^(1/2)/x)*Pi^(1/2)/b^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5346, 5298, 2204, 2205} \[ \frac {\sqrt {\pi } e^{-a} \text {Erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {\sqrt {\pi } e^a \text {Erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x^2]/x^2,x]

[Out]

(Sqrt[Pi]*Erf[Sqrt[b]/x])/(4*Sqrt[b]*E^a) - (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(4*Sqrt[b])

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5298

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5346

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subst[Int[(a + b*Sinh[c + d/
x^n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[p] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sinh \left (a+\frac {b}{x^2}\right )}{x^2} \, dx &=-\operatorname {Subst}\left (\int \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int e^{-a-b x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} \operatorname {Subst}\left (\int e^{a+b x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}-\frac {e^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b}}{x}\right )}{4 \sqrt {b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 50, normalized size = 0.88 \[ \frac {\sqrt {\pi } \left ((\cosh (a)-\sinh (a)) \text {erf}\left (\frac {\sqrt {b}}{x}\right )-(\sinh (a)+\cosh (a)) \text {erfi}\left (\frac {\sqrt {b}}{x}\right )\right )}{4 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b/x^2]/x^2,x]

[Out]

(Sqrt[Pi]*(Erf[Sqrt[b]/x]*(Cosh[a] - Sinh[a]) - Erfi[Sqrt[b]/x]*(Cosh[a] + Sinh[a])))/(4*Sqrt[b])

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 52, normalized size = 0.91 \[ \frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \operatorname {erf}\left (\frac {\sqrt {-b}}{x}\right ) + \sqrt {\pi } \sqrt {b} {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \operatorname {erf}\left (\frac {\sqrt {b}}{x}\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(pi)*sqrt(-b)*(cosh(a) + sinh(a))*erf(sqrt(-b)/x) + sqrt(pi)*sqrt(b)*(cosh(a) - sinh(a))*erf(sqrt(b)/
x))/b

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(a + b/x^2)/x^2, x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 44, normalized size = 0.77 \[ \frac {\erf \left (\frac {\sqrt {b}}{x}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{4 \sqrt {b}}-\frac {{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\frac {\sqrt {-b}}{x}\right )}{4 \sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*erf(b^(1/2)/x)*Pi^(1/2)*exp(-a)/b^(1/2)-1/4*exp(a)*Pi^(1/2)/(-b)^(1/2)*erf((-b)^(1/2)/x)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 62, normalized size = 1.09 \[ -\frac {1}{2} \, b {\left (\frac {e^{\left (-a\right )} \Gamma \left (\frac {3}{2}, \frac {b}{x^{2}}\right )}{x^{3} \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}} + \frac {e^{a} \Gamma \left (\frac {3}{2}, -\frac {b}{x^{2}}\right )}{x^{3} \left (-\frac {b}{x^{2}}\right )^{\frac {3}{2}}}\right )} - \frac {\sinh \left (a + \frac {b}{x^{2}}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {sinh}\left (a+\frac {b}{x^2}\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b/x^2)/x^2,x)

[Out]

int(sinh(a + b/x^2)/x^2, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\left (a + \frac {b}{x^{2}} \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b/x**2)/x**2, x)

________________________________________________________________________________________