\(\int x \sinh (a+b x) \text {Shi}(c+d x) \, dx\) [63]

Optimal result

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

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

Mathematica [A] (verified)

Time = 2.73 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.79 \[ \int x \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {-\frac {(b c+d) e^{a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )}{d}+\frac {e^{-a-c-(b+d) x} \left (b d \left (b \left (-1+e^{2 (a+b x)}\right )+d \left (1+e^{2 (a+b x)}\right )\right )-(b c-d) \left (b^2-d^2\right ) e^{\frac {(b+d) (c+d x)}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )}{(b-d) d (b+d)}-\frac {e^{-a} \left (b d e^c \left (\frac {e^{(-b+d) x}}{-b+d}+\frac {e^{2 a+(b+d) x}}{b+d}\right )+(-b c+d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )-(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+4 (b x \cosh (a+b x)-\sinh (a+b x)) \text {Shi}(c+d x)}{4 b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.63 (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 = {7096, 7100, 5993, 2009, 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.

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

\(\Big \downarrow \) 7096

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

\(\Big \downarrow \) 7100

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

\(\Big \downarrow \) 5993

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(x*Cosh[a + b*x]*SinhIntegral[c + d*x])/b - (d*(-1/2*Cosh[a - c + (b - d)* 
x]/((b - d)*d) + Cosh[a + c + (b + d)*x]/(2*d*(b + d)) + (c*CoshIntegral[( 
c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*d^2) - (c*CoshIntegral[(c* 
(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*d^2) + (c*Cosh[a - (b*c)/d]* 
SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*d^2) - (c*Cosh[a - (b*c)/d]*Si 
nhIntegral[(c*(b + d))/d + (b + d)*x])/(2*d^2)))/b - ((Sinh[a + b*x]*SinhI 
ntegral[c + d*x])/b - (d*(-1/2*(Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d)) 
/d + (b - d)*x])/d + (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + 
d)*x])/(2*d) - (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x]) 
/(2*d) + (Sinh[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 5993

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

rule 7096

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

rule 7100

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

rule 7293

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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