Integrand size = 13, antiderivative size = 153 \[ \int \cosh (a+b x) \text {Chi}(c+d x) \, dx=-\frac {\text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b}-\frac {\text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b}+\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \]
-1/2*cosh(a-b*c/d)*Shi(c*(b-d)/d+(b-d)*x)/b-1/2*cosh(a-b*c/d)*Shi(c*(b+d)/ d+(b+d)*x)/b-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+Chi(d*x+c)*sinh(b*x+a)/b
Time = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \cosh (a+b x) \text {Chi}(c+d x) \, dx=\frac {e^{-a-\frac {b c}{d}} \left (e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )+4 e^{a+\frac {b c}{d}} \text {Chi}(c+d x) \sinh (a+b x)\right )}{4 b} \]
(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)*ExpIntegralE i[-(((b + d)*(c + d*x))/d)] - E^(2*a)*ExpIntegralEi[((b + d)*(c + d*x))/d] + 4*E^(a + (b*c)/d)*CoshIntegral[c + d*x]*Sinh[a + b*x]))/(4*b)
Time = 0.52 (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 = {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.
\(\displaystyle \int \cosh (a+b x) \text {Chi}(c+d x) \, dx\) |
\(\Big \downarrow \) 7095 |
\(\displaystyle \frac {\sinh (a+b x) \text {Chi}(c+d x)}{b}-\frac {d \int \frac {\cosh (c+d x) \sinh (a+b x)}{c+d x}dx}{b}\) |
\(\Big \downarrow \) 5995 |
\(\displaystyle \frac {\sinh (a+b x) \text {Chi}(c+d x)}{b}-\frac {d \int \left (\frac {\sinh (a-c+(b-d) x)}{2 (c+d x)}+\frac {\sinh (a+c+(b+d) x)}{2 (c+d x)}\right )dx}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sinh (a+b x) \text {Chi}(c+d x)}{b}-\frac {d \left (\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}\right )}{b}\) |
(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
3.2.35.3.1 Defintions of rubi rules used
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]
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]
\[\int \operatorname {Chi}\left (d x +c \right ) \cosh \left (b x +a \right )d x\]
\[ \int \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int { {\rm Chi}\left (d x + c\right ) \cosh \left (b x + a\right ) \,d x } \]
\[ \int \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int \cosh {\left (a + b x \right )} \operatorname {Chi}\left (c + d x\right )\, dx \]
\[ \int \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int { {\rm Chi}\left (d x + c\right ) \cosh \left (b x + a\right ) \,d x } \]
\[ \int \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int { {\rm Chi}\left (d x + c\right ) \cosh \left (b x + a\right ) \,d x } \]
Timed out. \[ \int \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int \mathrm {coshint}\left (c+d\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]