\(\int \sinh (a+b x) \text {Shi}(c+d x) \, dx\) [64]

Optimal result

Integrand size = 13, antiderivative size = 153 \[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {\text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b}-\frac {\text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac {\cosh (a+b x) \text {Shi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {e^{-a-\frac {b c}{d}} \left (-e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )+4 e^{a+\frac {b c}{d}} \cosh (a+b x) \text {Shi}(c+d x)\right )}{4 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {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.

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

\(\Big \downarrow \) 7094

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

\(\Big \downarrow \) 5995

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(Cosh[a + b*x]*SinhIntegral[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
 

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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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