\(\int x \cosh (a+b x) \text {Shi}(c+d x) \, dx\) [66]

Optimal result

Integrand size = 14, antiderivative size = 371 \[ \int x \cosh (a+b x) \text {Shi}(c+d x) \, dx=-\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}-\frac {\text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b^2}+\frac {\text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b^2}+\frac {\sinh (a-c+(b-d) x)}{2 b (b-d)}-\frac {\sinh (a+c+(b+d) x)}{2 b (b+d)}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}-\frac {\cosh (a+b x) \text {Shi}(c+d x)}{b^2}+\frac {x \sinh (a+b x) \text {Shi}(c+d x)}{b}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d} \] Output:

-1/2*c*cosh(a-b*c/d)*Chi(c*(b-d)/d+(b-d)*x)/b/d+1/2*c*cosh(a-b*c/d)*Chi(c* 
(b+d)/d+(b+d)*x)/b/d-1/2*Chi(c*(b-d)/d+(b-d)*x)*sinh(a-b*c/d)/b^2+1/2*Chi( 
c*(b+d)/d+(b+d)*x)*sinh(a-b*c/d)/b^2+1/2*sinh(a-c+(b-d)*x)/b/(b-d)-1/2*sin 
h(a+c+(b+d)*x)/b/(b+d)-1/2*cosh(a-b*c/d)*Shi(c*(b-d)/d+(b-d)*x)/b^2-1/2*c* 
sinh(a-b*c/d)*Shi(c*(b-d)/d+(b-d)*x)/b/d-cosh(b*x+a)*Shi(d*x+c)/b^2+x*sinh 
(b*x+a)*Shi(d*x+c)/b+1/2*cosh(a-b*c/d)*Shi(c*(b+d)/d+(b+d)*x)/b^2+1/2*c*si 
nh(a-b*c/d)*Shi(c*(b+d)/d+(b+d)*x)/b/d
 

Mathematica [A] (verified)

Time = 2.91 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.72 \[ \int x \cosh (a+b x) \text {Shi}(c+d x) \, dx=\frac {\frac {e^{-a} \left (b d e^{-c} \left (\frac {e^{-((b+d) x)}}{b+d}+\frac {e^{2 a+b x-d x}}{b-d}\right )-(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )+(b c-d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+\frac {e^{-a} \left (b d e^c \left (\frac {e^{(-b+d) x}}{-b+d}-\frac {e^{2 a+(b+d) x}}{b+d}\right )+(-b c+d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+4 (-\cosh (a+b x)+b x \sinh (a+b x)) \text {Shi}(c+d x)}{4 b^2} \] Input:

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

Output:

(((b*d*(1/((b + d)*E^((b + d)*x)) + E^(2*a + b*x - d*x)/(b - d)))/E^c - (b 
*c + d)*E^(2*a - (b*c)/d)*ExpIntegralEi[((b - d)*(c + d*x))/d] + (b*c - d) 
*E^((b*c)/d)*ExpIntegralEi[-(((b + d)*(c + d*x))/d)])/(d*E^a) + (b*d*E^c*( 
E^((-b + d)*x)/(-b + d) - E^(2*a + (b + d)*x)/(b + d)) + (-(b*c) + d)*E^(( 
b*c)/d)*ExpIntegralEi[-(((b - d)*(c + d*x))/d)] + (b*c + d)*E^(2*a - (b*c) 
/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(d*E^a) + 4*(-Cosh[a + b*x] + b* 
x*Sinh[a + b*x])*SinhIntegral[c + d*x])/(4*b^2)
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {7102, 6176, 2009, 7094, 5995, 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.

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

\(\Big \downarrow \) 7102

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

\(\Big \downarrow \) 6176

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7094

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

\(\Big \downarrow \) 5995

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(x*Sinh[a + b*x]*SinhIntegral[c + d*x])/b - (d*((c*Cosh[a - (b*c)/d]*CoshI 
ntegral[(c*(b - d))/d + (b - d)*x])/(2*d^2) - (c*Cosh[a - (b*c)/d]*CoshInt 
egral[(c*(b + d))/d + (b + d)*x])/(2*d^2) - Sinh[a - c + (b - d)*x]/(2*(b 
- d)*d) + Sinh[a + c + (b + d)*x]/(2*d*(b + d)) + (c*Sinh[a - (b*c)/d]*Sin 
hIntegral[(c*(b - d))/d + (b - d)*x])/(2*d^2) - (c*Sinh[a - (b*c)/d]*SinhI 
ntegral[(c*(b + d))/d + (b + d)*x])/(2*d^2)))/b - ((Cosh[a + b*x]*SinhInte 
gral[c + d*x])/b - (d*(-1/2*(CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[ 
a - (b*c)/d])/d + (CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/ 
d])/(2*d) - (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2 
*d) + (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*d)))/ 
b)/b
 

Defintions of rubi rules used

rule 2009

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

rule 5995

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

rule 6176

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

\[\int x \cosh \left (b x +a \right ) \operatorname {Shi}\left (d x +c \right )d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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