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

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

(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*Sin 
hIntegral[2*(a + b*x)])/(4*b^2)
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {7102, 7094, 5971, 27, 3042, 26, 3779, 7293, 2009}

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.

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

\(\Big \downarrow \) 7102

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

\(\Big \downarrow \) 7094

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

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]
 

rule 7293

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

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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