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

Optimal. Leaf size=153 \[ \frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b}+\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}+\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.242279, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {6546, 5470, 3303, 3298, 3301} \[ \frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}-\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b}+\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b}+\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b} \]

Antiderivative was successfully verified.

[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 6546

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]

Rule 5470

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 3303

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 3298

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 3301

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]

Rubi steps

\begin{align*} \int \cosh (a+b x) \text{Shi}(c+d x) \, dx &=\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]  time = 0.581101, size = 209, normalized size = 1.37 \[ \frac{2 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (-\frac{(b-d) (c+d x)}{d}\right )-2 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{(b+d) (c+d x)}{d}\right )+4 \sinh (a+b x) \text{Shi}(c+d x)+\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right )-2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b+d) (c+d x)}{d}\right )-\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right )+\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right )+\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.154, size = 0, normalized size = 0. \begin{align*} \int \cosh \left ( bx+a \right ){\it Shi} \left ( dx+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (d x + c\right ) \cosh \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x + a\right ) \operatorname{Shi}\left (d x + c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (a + b x \right )} \operatorname{Shi}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (d x + c\right ) \cosh \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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