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

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

(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)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {7102, 27, 3042, 25, 3115, 24, 7094, 27, 5971, 27, 3042, 26, 3779}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7102

\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\int \frac {\sinh ^2(b x)}{b}dx+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\frac {\int \sinh ^2(b x)dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\frac {\int -\sin (i b x)^2dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {\int \sin (i b x)^2dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {\frac {\int 1dx}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

\(\Big \downarrow \) 7094

\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b x}dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\cosh (b x) \sinh (b x)}{x}dx}{b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{2 x}dx}{b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int -\frac {i \sin (2 i b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}+\frac {i \int \frac {\sin (2 i b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {x \text {Shi}(b x) \sinh (b x)}{b}-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b}}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 24

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 5971

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 7094

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

rule 7102

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] + (-Simp[d/b   Int[(e + f*x)^m*Sinh[a + b*x]*(Sinh[c + d*x]/( 
c + d*x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*Sinh 
Integral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.79 (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	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b x \sinh \left (b x \right )-\cosh \left (b x \right )\right )-\frac {\cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b x}{2}+\frac {\operatorname {Shi}\left (2 b x \right )}{2}}{b^{2}}\)	\(46\)

Input:

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

Output:

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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \cosh (b x) \text {Shi}(b x) \, dx=\int \cosh \left (b x \right ) \mathit {shi} \left (b x \right ) x d x \] Input:

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

Output:

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