\(\int x \cosh (b x) \text {Shi}(b x) \, dx\) [50]

Optimal result

Integrand size = 10, antiderivative size = 62 \[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\frac {x}{2 b}-\frac {\cosh (b x) \sinh (b x)}{2 b^2}-\frac {\cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x \sinh (b x) \text {Shi}(b x)}{b}+\frac {\text {Shi}(2 b x)}{2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6683, 12, 2715, 8, 6675, 5556, 3379} \[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\frac {\text {Shi}(2 b x)}{2 b^2}-\frac {\text {Shi}(b x) \cosh (b x)}{b^2}-\frac {\sinh (b x) \cosh (b x)}{2 b^2}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {x}{2 b} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3379

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]

Rule 5556

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

Rule 6675

Int[Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cosh[a + b*x]*(SinhIntegral[c
+ d*x]/b), x] - Dist[d/b, Int[Cosh[a + b*x]*(Sinh[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 6683

Int[Cosh[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e
+ f*x)^m*Sinh[a + b*x]*(SinhIntegral[c + d*x]/b), x] + (-Dist[d/b, Int[(e + f*x)^m*Sinh[a + b*x]*(Sinh[c + d*
x]/(c + d*x)), x], x] - Dist[f*(m/b), Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x], x]) /; Fr
eeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\frac {2 b x-\sinh (2 b x)+4 (-\cosh (b x)+b x \sinh (b x)) \text {Shi}(b x)+2 \text {Shi}(2 b x)}{4 b^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b x}{2}+\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b^{2}}\)	\(46\)
default	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b x}{2}+\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b^{2}}\)	\(46\)

[In]

int(x*cosh(b*x)*Shi(b*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int { x {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]

[In]

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

[Out]

integral(x*cosh(b*x)*sinh_integral(b*x), x)

Sympy [F]

\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int x \cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]

[In]

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

[Out]

Integral(x*cosh(b*x)*Shi(b*x), x)

Maxima [F]

\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int { x {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]

[In]

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

[Out]

integrate(x*Shi(b*x)*cosh(b*x), x)

Giac [F]

\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int { x {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]

[In]

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

[Out]

integrate(x*Shi(b*x)*cosh(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int x\,\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]

[In]

int(x*sinhint(b*x)*cosh(b*x),x)

[Out]

int(x*sinhint(b*x)*cosh(b*x), x)