\(\int \cosh (a+b x) \text {Shi}(c+d x) \, dx\) [67]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6681, 5578, 3384, 3379, 3382} \[ \int \cosh (a+b x) \text {Shi}(c+d x) \, dx=\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}+\frac {\sinh (a+b x) \text {Shi}(c+d x)}{b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b} \]

[In]

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

[Out]

(Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d)
)/d + (b + d)*x])/(2*b) + (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) + (Sinh[a + b*x]*S
inhIntegral[c + d*x])/b - (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5578

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 6681

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

Rubi steps \begin{align*} \text {integral}& = \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} \\ & = \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} \\ & = \frac {\sinh (a+b x) \text {Shi}(c+d x)}{b}+\frac {d \int \frac {\cosh (a-c+(b-d) x)}{c+d x} \, dx}{2 b}-\frac {d \int \frac {\cosh (a+c+(b+d) x)}{c+d x} \, dx}{2 b} \\ & = \frac {\sinh (a+b x) \text {Shi}(c+d x)}{b}+\frac {\left (d \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}-\frac {\left (d \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}+\frac {\left (d \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}-\frac {\left (d \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b} \\ & = \frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac {\sinh (a+b x) \text {Shi}(c+d x)}{b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \cosh (a+b x) \text {Shi}(c+d x) \, dx=\frac {e^{-a-\frac {b c}{d}} \left (e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )-e^{\frac {2 b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )+4 e^{a+\frac {b c}{d}} \sinh (a+b x) \text {Shi}(c+d x)\right )}{4 b} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sinhint(c + d*x)*cosh(a + b*x),x)

[Out]

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