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

Optimal. Leaf size=371 \[ -\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}+\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}-\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b^2}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{c \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{c \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}-\frac{c \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{x \sinh (a+b x) \text{Shi}(c+d x)}{b}+\frac{c \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}+\frac{\sinh (a+x (b-d)-c)}{2 b (b-d)}-\frac{\sinh (a+x (b+d)+c)}{2 b (b+d)} \]

[Out]

-(c*Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (c*Cosh[a - (b*c)/d]*CoshIntegral[(c*
(b + d))/d + (b + d)*x])/(2*b*d) - (CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b^2) + (Cosh
Integral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b^2) + Sinh[a - c + (b - d)*x]/(2*b*(b - d)) - Sinh[
a + c + (b + d)*x]/(2*b*(b + d)) - (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) - (c*Si
nh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (Cosh[a + b*x]*SinhIntegral[c + d*x])/b^2 +
(x*Sinh[a + b*x]*SinhIntegral[c + d*x])/b + (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^
2) + (c*Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.31689, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6548, 5642, 6742, 2637, 3303, 3298, 3301, 6540, 5472} \[ -\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}+\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}-\frac{\cosh (a+b x) \text{Shi}(c+d x)}{b^2}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{c \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{c \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}-\frac{c \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{x \sinh (a+b x) \text{Shi}(c+d x)}{b}+\frac{c \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}+\frac{\sinh (a+x (b-d)-c)}{2 b (b-d)}-\frac{\sinh (a+x (b+d)+c)}{2 b (b+d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (c*Cosh[a - (b*c)/d]*CoshIntegral[(c*
(b + d))/d + (b + d)*x])/(2*b*d) - (CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b^2) + (Cosh
Integral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b^2) + Sinh[a - c + (b - d)*x]/(2*b*(b - d)) - Sinh[
a + c + (b + d)*x]/(2*b*(b + d)) - (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) - (c*Si
nh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (Cosh[a + b*x]*SinhIntegral[c + d*x])/b^2 +
(x*Sinh[a + b*x]*SinhIntegral[c + d*x])/b + (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^
2) + (c*Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)

Rule 6548

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

Rule 5642

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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]

Rule 6540

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

Rule 5472

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] && I
GtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [B]  time = 8.20198, size = 887, normalized size = 2.39 \[ \frac{4 d x \sinh (a+b x) \text{Shi}(c+d x) b^3-c \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right ) b^3-c \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right ) b^3+2 c \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b+d) (c+d x)}{d}\right ) b^3-c \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right ) b^3+c \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right ) b^3+2 d \sinh (a-c+b x-d x) b^2-2 d \sinh (a+c+(b+d) x) b^2-4 d \cosh (a+b x) \text{Shi}(c+d x) b^2-d \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right ) b^2-d \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right ) b^2+2 d \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b+d) (c+d x)}{d}\right ) b^2+d \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right ) b^2-d \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right ) b^2+2 d^2 \sinh (a-c+b x-d x) b+2 d^2 \sinh (a+c+(b+d) x) b-4 d^3 x \sinh (a+b x) \text{Shi}(c+d x) b+c d^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right ) b+c d^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right ) b-2 c d^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b+d) (c+d x)}{d}\right ) b+c d^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right ) b-c d^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right ) b-2 \left (b^2-d^2\right ) \text{Chi}\left (-\frac{(b-d) (c+d x)}{d}\right ) \left (b c \cosh \left (a-\frac{b c}{d}\right )+d \sinh \left (a-\frac{b c}{d}\right )\right )+2 \left (b^2-d^2\right ) \text{Chi}\left (\frac{(b+d) (c+d x)}{d}\right ) \left (b c \cosh \left (a-\frac{b c}{d}\right )+d \sinh \left (a-\frac{b c}{d}\right )\right )+4 d^3 \cosh (a+b x) \text{Shi}(c+d x)+d^3 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right )+d^3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b-d) (c+d x)}{d}\right )-2 d^3 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{(b+d) (c+d x)}{d}\right )-d^3 \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right )+d^3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (-\frac{b c}{d}+c-b x+d x\right )}{4 b^2 (b-d) d (b+d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^2 - d^2)*CoshIntegral[-(((b - d)*(c + d*x))/d)]*(b*c*Cosh[a - (b*c)/d] + d*Sinh[a - (b*c)/d]) + 2*(b^2
- d^2)*CoshIntegral[((b + d)*(c + d*x))/d]*(b*c*Cosh[a - (b*c)/d] + d*Sinh[a - (b*c)/d]) + 2*b^2*d*Sinh[a - c
+ b*x - d*x] + 2*b*d^2*Sinh[a - c + b*x - d*x] - 2*b^2*d*Sinh[a + c + (b + d)*x] + 2*b*d^2*Sinh[a + c + (b + d
)*x] - 4*b^2*d*Cosh[a + b*x]*SinhIntegral[c + d*x] + 4*d^3*Cosh[a + b*x]*SinhIntegral[c + d*x] + 4*b^3*d*x*Sin
h[a + b*x]*SinhIntegral[c + d*x] - 4*b*d^3*x*Sinh[a + b*x]*SinhIntegral[c + d*x] - b^3*c*Cosh[a - (b*c)/d]*Sin
hIntegral[((b - d)*(c + d*x))/d] - b^2*d*Cosh[a - (b*c)/d]*SinhIntegral[((b - d)*(c + d*x))/d] + b*c*d^2*Cosh[
a - (b*c)/d]*SinhIntegral[((b - d)*(c + d*x))/d] + d^3*Cosh[a - (b*c)/d]*SinhIntegral[((b - d)*(c + d*x))/d] -
b^3*c*Sinh[a - (b*c)/d]*SinhIntegral[((b - d)*(c + d*x))/d] - b^2*d*Sinh[a - (b*c)/d]*SinhIntegral[((b - d)*(
c + d*x))/d] + b*c*d^2*Sinh[a - (b*c)/d]*SinhIntegral[((b - d)*(c + d*x))/d] + d^3*Sinh[a - (b*c)/d]*SinhInteg
ral[((b - d)*(c + d*x))/d] + 2*b^2*d*Cosh[a - (b*c)/d]*SinhIntegral[((b + d)*(c + d*x))/d] - 2*d^3*Cosh[a - (b
*c)/d]*SinhIntegral[((b + d)*(c + d*x))/d] + 2*b^3*c*Sinh[a - (b*c)/d]*SinhIntegral[((b + d)*(c + d*x))/d] - 2
*b*c*d^2*Sinh[a - (b*c)/d]*SinhIntegral[((b + d)*(c + d*x))/d] - b^3*c*Cosh[a - (b*c)/d]*SinhIntegral[c - (b*c
)/d - b*x + d*x] + b^2*d*Cosh[a - (b*c)/d]*SinhIntegral[c - (b*c)/d - b*x + d*x] + b*c*d^2*Cosh[a - (b*c)/d]*S
inhIntegral[c - (b*c)/d - b*x + d*x] - d^3*Cosh[a - (b*c)/d]*SinhIntegral[c - (b*c)/d - b*x + d*x] + b^3*c*Sin
h[a - (b*c)/d]*SinhIntegral[c - (b*c)/d - b*x + d*x] - b^2*d*Sinh[a - (b*c)/d]*SinhIntegral[c - (b*c)/d - b*x
+ d*x] - b*c*d^2*Sinh[a - (b*c)/d]*SinhIntegral[c - (b*c)/d - b*x + d*x] + d^3*Sinh[a - (b*c)/d]*SinhIntegral[
c - (b*c)/d - b*x + d*x])/(4*b^2*(b - d)*d*(b + d))

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Maple [F]  time = 0.303, size = 0, normalized size = 0. \begin{align*} \int x\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(x*cosh(b*x+a)*Shi(d*x+c),x)

[Out]

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

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

[Out]

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

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

[Out]

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

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

[Out]

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

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

[Out]

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