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

Optimal. Leaf size=25 \[ -\frac{1}{2} \sinh (a) \text{Chi}\left (\frac{b}{x^2}\right )-\frac{1}{2} \cosh (a) \text{Shi}\left (\frac{b}{x^2}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0329483, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5318, 5317, 5316} \[ -\frac{1}{2} \sinh (a) \text{Chi}\left (\frac{b}{x^2}\right )-\frac{1}{2} \cosh (a) \text{Shi}\left (\frac{b}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b/x^2]/x,x]

[Out]

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

Rule 5318

Int[Sinh[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Sinh[c], Int[Cosh[d*x^n]/x, x], x] + Dist[Cosh[c], In
t[Sinh[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 5317

Int[Cosh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CoshIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 5316

Int[Sinh[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinhIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0139663, size = 25, normalized size = 1. \[ \frac{1}{2} \left (\sinh (a) \left (-\text{Chi}\left (\frac{b}{x^2}\right )\right )-\cosh (a) \text{Shi}\left (\frac{b}{x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b/x^2]/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 27, normalized size = 1.1 \begin{align*} -{\frac{{{\rm e}^{-a}}}{4}{\it Ei} \left ( 1,{\frac{b}{{x}^{2}}} \right ) }+{\frac{{{\rm e}^{a}}}{4}{\it Ei} \left ( 1,-{\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,x)

[Out]

-1/4*exp(-a)*Ei(1,b/x^2)+1/4*exp(a)*Ei(1,-b/x^2)

________________________________________________________________________________________

Maxima [A]  time = 1.38342, size = 32, normalized size = 1.28 \begin{align*} \frac{1}{4} \,{\rm Ei}\left (-\frac{b}{x^{2}}\right ) e^{\left (-a\right )} - \frac{1}{4} \,{\rm Ei}\left (\frac{b}{x^{2}}\right ) e^{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.70199, size = 105, normalized size = 4.2 \begin{align*} -\frac{1}{4} \,{\left ({\rm Ei}\left (\frac{b}{x^{2}}\right ) -{\rm Ei}\left (-\frac{b}{x^{2}}\right )\right )} \cosh \left (a\right ) - \frac{1}{4} \,{\left ({\rm Ei}\left (\frac{b}{x^{2}}\right ) +{\rm Ei}\left (-\frac{b}{x^{2}}\right )\right )} \sinh \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(Ei(b/x^2) - Ei(-b/x^2))*cosh(a) - 1/4*(Ei(b/x^2) + Ei(-b/x^2))*sinh(a)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b/x**2)/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(a + b/x^2)/x, x)