3.2.34 \(\int x \cosh (a+b x) \text {Chi}(c+d x) \, dx\) [134]

3.2.34.1 Optimal result

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

output

1/2*Chi(c*(b-d)/d+(b-d)*x)*cosh(a-b*c/d)/b^2+1/2*Chi(c*(b+d)/d+(b+d)*x)*co 
sh(a-b*c/d)/b^2-Chi(d*x+c)*cosh(b*x+a)/b^2-1/2*cosh(a-c+(b-d)*x)/b/(b-d)-1 
/2*cosh(a+c+(b+d)*x)/b/(b+d)+1/2*c*cosh(a-b*c/d)*Shi(c*(b-d)/d+(b-d)*x)/b/ 
d+1/2*c*cosh(a-b*c/d)*Shi(c*(b+d)/d+(b+d)*x)/b/d+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*S 
hi(c*(b-d)/d+(b-d)*x)*sinh(a-b*c/d)/b^2+1/2*Shi(c*(b+d)/d+(b+d)*x)*sinh(a- 
b*c/d)/b^2+x*Chi(d*x+c)*sinh(b*x+a)/b
 

3.2.34.2 Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.73 \[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\frac {\frac {e^{-a} \left (b d e^{-c} \left (-\frac {e^{-((b+d) x)}}{b+d}-\frac {e^{2 a+b x-d x}}{b-d}\right )+(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )-(b c-d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+\frac {e^{-a} \left (b d e^c \left (\frac {e^{(-b+d) x}}{-b+d}-\frac {e^{2 a+(b+d) x}}{b+d}\right )+(-b c+d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+4 \text {Chi}(c+d x) (-\cosh (a+b x)+b x \sinh (a+b x))}{4 b^2} \]

input

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

3.2.34.3 Rubi [A] (verified)

Time = 1.47 (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 = {7097, 7101, 5994, 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.

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

\(\Big \downarrow \) 7097

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

\(\Big \downarrow \) 7101

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

\(\Big \downarrow \) 5994

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

(x*CoshIntegral[c + d*x]*Sinh[a + b*x])/b - (d*(Cosh[a - c + (b - d)*x]/(2 
*(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]*Sin 
hIntegral[(c*(b - d))/d + (b - d)*x])/(2*d^2) - (c*Cosh[a - (b*c)/d]*SinhI 
ntegral[(c*(b + d))/d + (b + d)*x])/(2*d^2)))/b - ((Cosh[a + b*x]*CoshInte 
gral[c + d*x])/b - (d*((Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b 
- d)*x])/(2*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
 

3.2.34.3.1 Defintions of rubi rules used

rule 2009

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

rule 5994

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

rule 7097

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

rule 7101

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

3.2.34.4 Maple [F]

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

input

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

output

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

3.2.34.5 Fricas [F]

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

input

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

output

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

3.2.34.6 Sympy [F]

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

input

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

output

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

3.2.34.7 Maxima [F]

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

input

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

output

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

3.2.34.8 Giac [F]

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

input

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

output

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

3.2.34.9 Mupad [F(-1)]

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

input

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

output

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