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

Optimal. Leaf size=67 \[ -\frac{1}{2} \sqrt{\pi } e^{-a} \sqrt{b} \text{Erf}\left (\frac{\sqrt{b}}{x}\right )-\frac{1}{2} \sqrt{\pi } e^a \sqrt{b} \text{Erfi}\left (\frac{\sqrt{b}}{x}\right )+x \sinh \left (a+\frac{b}{x^2}\right ) \]

[Out]

-(Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x])/(2*E^a) - (Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/2 + x*Sinh[a + b/x^2]

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Rubi [A]  time = 0.0436817, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5302, 5326, 5299, 2204, 2205} \[ -\frac{1}{2} \sqrt{\pi } e^{-a} \sqrt{b} \text{Erf}\left (\frac{\sqrt{b}}{x}\right )-\frac{1}{2} \sqrt{\pi } e^a \sqrt{b} \text{Erfi}\left (\frac{\sqrt{b}}{x}\right )+x \sinh \left (a+\frac{b}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x^2],x]

[Out]

-(Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x])/(2*E^a) - (Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/2 + x*Sinh[a + b/x^2]

Rule 5302

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

Rule 5326

Int[((e_.)*(x_))^(m_)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[((e*x)^(m + 1)*Sinh[c + d*x^n])/(e*(m +
1)), x] - Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[
n, 0] && LtQ[m, -1]

Rule 5299

Int[Cosh[(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 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]

Rubi steps

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

Mathematica [A]  time = 0.0669879, size = 70, normalized size = 1.04 \[ -\frac{1}{2} \sqrt{\pi } \sqrt{b} \left ((\cosh (a)-\sinh (a)) \text{Erf}\left (\frac{\sqrt{b}}{x}\right )+(\sinh (a)+\cosh (a)) \text{Erfi}\left (\frac{\sqrt{b}}{x}\right )\right )+x \sinh (a) \cosh \left (\frac{b}{x^2}\right )+x \cosh (a) \sinh \left (\frac{b}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b/x^2],x]

[Out]

x*Cosh[b/x^2]*Sinh[a] - (Sqrt[b]*Sqrt[Pi]*(Erf[Sqrt[b]/x]*(Cosh[a] - Sinh[a]) + Erfi[Sqrt[b]/x]*(Cosh[a] + Sin
h[a])))/2 + x*Cosh[a]*Sinh[b/x^2]

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Maple [A]  time = 0.031, size = 70, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{-a}}\sqrt{\pi }}{2}\sqrt{b}{\it Erf} \left ({\frac{1}{x}\sqrt{b}} \right ) }-{\frac{{{\rm e}^{-a}}x}{2}{{\rm e}^{-{\frac{b}{{x}^{2}}}}}}+{\frac{{{\rm e}^{a}}x}{2}{{\rm e}^{{\frac{b}{{x}^{2}}}}}}-{\frac{{{\rm e}^{a}}b\sqrt{\pi }}{2}{\it Erf} \left ({\frac{1}{x}\sqrt{-b}} \right ){\frac{1}{\sqrt{-b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.2211, size = 96, normalized size = 1.43 \begin{align*} -\frac{1}{2} \, b{\left (\frac{\sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{\frac{b}{x^{2}}}\right ) - 1\right )} e^{\left (-a\right )}}{x \sqrt{\frac{b}{x^{2}}}} + \frac{\sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-\frac{b}{x^{2}}}\right ) - 1\right )} e^{a}}{x \sqrt{-\frac{b}{x^{2}}}}\right )} + x \sinh \left (a + \frac{b}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*(sqrt(pi)*(erf(sqrt(b/x^2)) - 1)*e^(-a)/(x*sqrt(b/x^2)) + sqrt(pi)*(erf(sqrt(-b/x^2)) - 1)*e^a/(x*sqrt(
-b/x^2))) + x*sinh(a + b/x^2)

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Fricas [B]  time = 1.78681, size = 606, normalized size = 9.04 \begin{align*} \frac{x \cosh \left (\frac{a x^{2} + b}{x^{2}}\right )^{2} + \sqrt{\pi }{\left (\cosh \left (a\right ) \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) + \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) +{\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )\right )} \sqrt{-b} \operatorname{erf}\left (\frac{\sqrt{-b}}{x}\right ) - \sqrt{\pi }{\left (\cosh \left (a\right ) \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) - \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (a\right ) +{\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )\right )} \sqrt{b} \operatorname{erf}\left (\frac{\sqrt{b}}{x}\right ) + 2 \, x \cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) \sinh \left (\frac{a x^{2} + b}{x^{2}}\right ) + x \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )^{2} - x}{2 \,{\left (\cosh \left (\frac{a x^{2} + b}{x^{2}}\right ) + \sinh \left (\frac{a x^{2} + b}{x^{2}}\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x*cosh((a*x^2 + b)/x^2)^2 + sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) + cosh((a*x^2 + b)/x^2)*sinh(a) + (co
sh(a) + sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(-b)*erf(sqrt(-b)/x) - sqrt(pi)*(cosh(a)*cosh((a*x^2 + b)/x^2) - c
osh((a*x^2 + b)/x^2)*sinh(a) + (cosh(a) - sinh(a))*sinh((a*x^2 + b)/x^2))*sqrt(b)*erf(sqrt(b)/x) + 2*x*cosh((a
*x^2 + b)/x^2)*sinh((a*x^2 + b)/x^2) + x*sinh((a*x^2 + b)/x^2)^2 - x)/(cosh((a*x^2 + b)/x^2) + sinh((a*x^2 + b
)/x^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (a + \frac{b}{x^{2}} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b/x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (a + \frac{b}{x^{2}}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(a + b/x^2), x)