3.4 \(\int x \text{Shi}(b x) \, dx\)

Optimal. Leaf size=35 \[ \frac{\sinh (b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(b x)-\frac{x \cosh (b x)}{2 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.026271, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6532, 12, 3296, 2637} \[ \frac{\sinh (b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(b x)-\frac{x \cosh (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x*SinhIntegral[b*x],x]

[Out]

-(x*Cosh[b*x])/(2*b) + Sinh[b*x]/(2*b^2) + (x^2*SinhIntegral[b*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 3296

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 2637

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

Rubi steps

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

Mathematica [A]  time = 0.0073354, size = 35, normalized size = 1. \[ \frac{\sinh (b x)}{2 b^2}+\frac{1}{2} x^2 \text{Shi}(b x)-\frac{x \cosh (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*SinhIntegral[b*x],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{b}^{2}{x}^{2}{\it Shi} \left ( bx \right ) }{2}}-{\frac{bx\cosh \left ( bx \right ) }{2}}+{\frac{\sinh \left ( bx \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Shi(b*x),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Shi(b*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*sinh_integral(b*x), x)

________________________________________________________________________________________

Sympy [A]  time = 0.774986, size = 29, normalized size = 0.83 \begin{align*} \frac{x^{2} \operatorname{Shi}{\left (b x \right )}}{2} - \frac{x \cosh{\left (b x \right )}}{2 b} + \frac{\sinh{\left (b x \right )}}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Shi(b*x), x)