3.8 \(\int \frac{\text{Shi}(b x)}{x^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{4} b^2 \text{Shi}(b x)-\frac{\text{Shi}(b x)}{2 x^2}-\frac{\sinh (b x)}{4 x^2}-\frac{b \cosh (b x)}{4 x} \]

[Out]

-(b*Cosh[b*x])/(4*x) - Sinh[b*x]/(4*x^2) + (b^2*SinhIntegral[b*x])/4 - SinhIntegral[b*x]/(2*x^2)

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Rubi [A]  time = 0.0710725, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6532, 12, 3297, 3298} \[ \frac{1}{4} b^2 \text{Shi}(b x)-\frac{\text{Shi}(b x)}{2 x^2}-\frac{\sinh (b x)}{4 x^2}-\frac{b \cosh (b x)}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[SinhIntegral[b*x]/x^3,x]

[Out]

-(b*Cosh[b*x])/(4*x) - Sinh[b*x]/(4*x^2) + (b^2*SinhIntegral[b*x])/4 - SinhIntegral[b*x]/(2*x^2)

Rule 6532

Int[((c_.) + (d_.)*(x_))^(m_.)*SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*SinhInte
gral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Sinh[a + b*x])/(a + b*x), x], x] /
; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{\text{Shi}(b x)}{x^3} \, dx &=-\frac{\text{Shi}(b x)}{2 x^2}+\frac{1}{2} b \int \frac{\sinh (b x)}{b x^3} \, dx\\ &=-\frac{\text{Shi}(b x)}{2 x^2}+\frac{1}{2} \int \frac{\sinh (b x)}{x^3} \, dx\\ &=-\frac{\sinh (b x)}{4 x^2}-\frac{\text{Shi}(b x)}{2 x^2}+\frac{1}{4} b \int \frac{\cosh (b x)}{x^2} \, dx\\ &=-\frac{b \cosh (b x)}{4 x}-\frac{\sinh (b x)}{4 x^2}-\frac{\text{Shi}(b x)}{2 x^2}+\frac{1}{4} b^2 \int \frac{\sinh (b x)}{x} \, dx\\ &=-\frac{b \cosh (b x)}{4 x}-\frac{\sinh (b x)}{4 x^2}+\frac{1}{4} b^2 \text{Shi}(b x)-\frac{\text{Shi}(b x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0128088, size = 46, normalized size = 1. \[ \frac{1}{4} b^2 \text{Shi}(b x)-\frac{\text{Shi}(b x)}{2 x^2}-\frac{\sinh (b x)}{4 x^2}-\frac{b \cosh (b x)}{4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[SinhIntegral[b*x]/x^3,x]

[Out]

-(b*Cosh[b*x])/(4*x) - Sinh[b*x]/(4*x^2) + (b^2*SinhIntegral[b*x])/4 - SinhIntegral[b*x]/(2*x^2)

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Maple [A]  time = 0.048, size = 48, normalized size = 1. \begin{align*}{b}^{2} \left ( -{\frac{{\it Shi} \left ( bx \right ) }{2\,{b}^{2}{x}^{2}}}-{\frac{\sinh \left ( bx \right ) }{4\,{b}^{2}{x}^{2}}}-{\frac{\cosh \left ( bx \right ) }{4\,bx}}+{\frac{{\it Shi} \left ( bx \right ) }{4}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Shi(b*x)/x^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(Shi(b*x)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{Shi}\left (b x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(sinh_integral(b*x)/x^3, x)

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Sympy [A]  time = 0.89274, size = 39, normalized size = 0.85 \begin{align*} \frac{b^{2} \operatorname{Shi}{\left (b x \right )}}{4} - \frac{b \cosh{\left (b x \right )}}{4 x} - \frac{\sinh{\left (b x \right )}}{4 x^{2}} - \frac{\operatorname{Shi}{\left (b x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x)/x**3,x)

[Out]

b**2*Shi(b*x)/4 - b*cosh(b*x)/(4*x) - sinh(b*x)/(4*x**2) - Shi(b*x)/(2*x**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(Shi(b*x)/x^3, x)