\(\int x \cosh (a+b x) \text {Shi}(a+b x) \, dx\) [60]

Optimal result

Integrand size = 14, antiderivative size = 109 \[ \int x \cosh (a+b x) \text {Shi}(a+b x) \, dx=\frac {x}{2 b}+\frac {a \text {Chi}(2 a+2 b x)}{2 b^2}-\frac {a \log (a+b x)}{2 b^2}-\frac {\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {\cosh (a+b x) \text {Shi}(a+b x)}{b^2}+\frac {x \sinh (a+b x) \text {Shi}(a+b x)}{b}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2} \]

[Out]

1/2*x/b+1/2*a*Chi(2*b*x+2*a)/b^2-1/2*a*ln(b*x+a)/b^2-cosh(b*x+a)*Shi(b*x+a)/b^2+1/2*Shi(2*b*x+2*a)/b^2-1/2*cos
h(b*x+a)*sinh(b*x+a)/b^2+x*Shi(b*x+a)*sinh(b*x+a)/b

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6683, 6874, 2715, 8, 3393, 3382, 6675, 5556, 12, 3379} \[ \int x \cosh (a+b x) \text {Shi}(a+b x) \, dx=\frac {a \text {Chi}(2 a+2 b x)}{2 b^2}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2}-\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b^2}-\frac {a \log (a+b x)}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {x}{2 b} \]

[In]

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

[Out]

x/(2*b) + (a*CoshIntegral[2*a + 2*b*x])/(2*b^2) - (a*Log[a + b*x])/(2*b^2) - (Cosh[a + b*x]*Sinh[a + b*x])/(2*
b^2) - (Cosh[a + b*x]*SinhIntegral[a + b*x])/b^2 + (x*Sinh[a + b*x]*SinhIntegral[a + b*x])/b + SinhIntegral[2*
a + 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 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int x \cosh (a+b x) \text {Shi}(a+b x) \, dx=\frac {2 b x+2 a \text {Chi}(2 (a+b x))-2 a \log (a+b x)-\sinh (2 (a+b x))+4 (-\cosh (a+b x)+b x \sinh (a+b x)) \text {Shi}(a+b x)+2 \text {Shi}(2 (a+b x))}{4 b^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+a \left (-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}\right )-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}+\frac {\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(97\)
default	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+a \left (-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}\right )-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}+\frac {\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(97\)

[In]

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

[Out]

1/b^2*(Shi(b*x+a)*(-a*sinh(b*x+a)+(b*x+a)*sinh(b*x+a)-cosh(b*x+a))+a*(-1/2*ln(b*x+a)+1/2*Chi(2*b*x+2*a))-1/2*c
osh(b*x+a)*sinh(b*x+a)+1/2*b*x+1/2*a+1/2*Shi(2*b*x+2*a))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x*cosh(a + b*x)*Shi(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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