\(\int x \text {Chi}(c+d x) \sinh (a+b x) \, dx\) [131]

Optimal result

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

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

Mathematica [A] (verified)

Time = 2.40 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.87 \[ \int x \text {Chi}(c+d x) \sinh (a+b x) \, dx=\frac {e^{-a-c-(b+d) x} \left (b d \left (d \left (-1+e^{2 (c+d x)}\right )+b \left (1+e^{2 (c+d x)}\right )\right )+(b c-d) \left (b^2-d^2\right ) e^{\frac {(b+d) (c+d x)}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+(b c-d) \left (b^2-d^2\right ) e^{\frac {(b+d) (c+d x)}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )+e^{a-\frac {c (b+d)}{d}} \left (-b d e^{\frac {b c}{d}+b x-d x} \left (b+d+b e^{2 (c+d x)}-d e^{2 (c+d x)}\right )+(b c+d) \left (b^2-d^2\right ) e^c \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )+(b c+d) \left (b^2-d^2\right ) e^c \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )+4 (b-d) d (b+d) \text {Chi}(c+d x) (b x \cosh (a+b x)-\sinh (a+b x))}{4 b^2 (b-d) d (b+d)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.80 (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 = {7103, 6177, 2009, 7095, 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.

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

\(\Big \downarrow \) 7103

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

\(\Big \downarrow \) 6177

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7095

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

\(\Big \downarrow \) 5995

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(x*Cosh[a + b*x]*CoshIntegral[c + d*x])/b - (d*(-1/2*(c*Cosh[a - (b*c)/d]* 
CoshIntegral[(c*(b - d))/d + (b - d)*x])/d^2 - (c*Cosh[a - (b*c)/d]*CoshIn 
tegral[(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]*Si 
nhIntegral[(c*(b - d))/d + (b - d)*x])/(2*d^2) - (c*Sinh[a - (b*c)/d]*Sinh 
Integral[(c*(b + d))/d + (b + d)*x])/(2*d^2)))/b - ((CoshIntegral[c + d*x] 
*Sinh[a + b*x])/b - (d*((CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - 
(b*c)/d])/(2*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 6177

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

rule 7095

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

rule 7103

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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