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

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

[Out]

Cosh[a - c + (b - d)*x]/(2*b*(b - d)) - Cosh[a + c + (b + d)*x]/(2*b*(b + d)) - (Cosh[a - (b*c)/d]*CoshIntegra
l[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) -
(c*CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b*d) + (c*CoshIntegral[(c*(b + d))/d + (b + d
)*x]*Sinh[a - (b*c)/d])/(2*b*d) - (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (Sin
h[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (x*Cosh[a + b*x]*SinhIntegral[c + d*x])/b -
(Sinh[a + b*x]*SinhIntegral[c + d*x])/b^2 + (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b
*d) + (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.963509, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {6542, 6742, 5618, 2638, 5472, 3303, 3298, 3301, 6546, 5470} \[ -\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}-\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b^2}+\frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{c \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{c \sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}-\frac{c \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{x \cosh (a+b x) \text{Shi}(c+d x)}{b}+\frac{c \cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}+\frac{\cosh (a+x (b-d)-c)}{2 b (b-d)}-\frac{\cosh (a+x (b+d)+c)}{2 b (b+d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Cosh[a - c + (b - d)*x]/(2*b*(b - d)) - Cosh[a + c + (b + d)*x]/(2*b*(b + d)) - (Cosh[a - (b*c)/d]*CoshIntegra
l[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) -
(c*CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b*d) + (c*CoshIntegral[(c*(b + d))/d + (b + d
)*x]*Sinh[a - (b*c)/d])/(2*b*d) - (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (Sin
h[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (x*Cosh[a + b*x]*SinhIntegral[c + d*x])/b -
(Sinh[a + b*x]*SinhIntegral[c + d*x])/b^2 + (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b
*d) + (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2)

Rule 6542

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

Rule 6742

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

Rule 5618

Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], x]))

Rule 2638

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

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 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]

Rubi steps

\begin{align*} \int x \sinh (a+b x) \text{Shi}(c+d x) \, dx &=\frac{x \cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\int \cosh (a+b x) \text{Shi}(c+d x) \, dx}{b}-\frac{d \int \frac{x \cosh (a+b x) \sinh (c+d x)}{c+d x} \, dx}{b}\\ &=\frac{x \cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b^2}+\frac{d \int \frac{\sinh (a+b x) \sinh (c+d x)}{c+d x} \, dx}{b^2}-\frac{d \int \left (\frac{\cosh (a+b x) \sinh (c+d x)}{d}-\frac{c \cosh (a+b x) \sinh (c+d x)}{d (c+d x)}\right ) \, dx}{b}\\ &=\frac{x \cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b^2}-\frac{\int \cosh (a+b x) \sinh (c+d x) \, dx}{b}+\frac{c \int \frac{\cosh (a+b x) \sinh (c+d x)}{c+d x} \, dx}{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^2}\\ &=\frac{x \cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b^2}-\frac{\int \left (-\frac{1}{2} \sinh (a-c+(b-d) x)+\frac{1}{2} \sinh (a+c+(b+d) x)\right ) \, dx}{b}+\frac{c \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}-\frac{d \int \frac{\cosh (a-c+(b-d) x)}{c+d x} \, dx}{2 b^2}+\frac{d \int \frac{\cosh (a+c+(b+d) x)}{c+d x} \, dx}{2 b^2}\\ &=\frac{x \cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b^2}+\frac{\int \sinh (a-c+(b-d) x) \, dx}{2 b}-\frac{\int \sinh (a+c+(b+d) x) \, dx}{2 b}-\frac{c \int \frac{\sinh (a-c+(b-d) x)}{c+d x} \, dx}{2 b}+\frac{c \int \frac{\sinh (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{\cosh \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{\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{\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{\sinh \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2}\\ &=\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 \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{c (b+d)}{d}+(b+d) x\right )}{2 b^2}-\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{x \cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh (a+b x) \text{Shi}(c+d x)}{b^2}+\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{\left (c \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}+\frac{\left (c \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}-\frac{\left (c \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}+\frac{\left (c \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}\\ &=\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 \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{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{x \cosh (a+b x) \text{Shi}(c+d x)}{b}-\frac{\sinh (a+b x) \text{Shi}(c+d x)}{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}\\ \end{align*}

Mathematica [B]  time = 17.3677, size = 887, normalized size = 2.39 \[ \frac{4 d x \cosh (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 \cosh \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 \cosh (a-c+b x-d x) b^2-2 d \cosh (a+c+(b+d) x) b^2-4 d \sinh (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 \sinh \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 \cosh (a-c+b x-d x) b+2 d^2 \cosh (a+c+(b+d) x) b-4 d^3 x \cosh (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 \cosh \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 (d \cosh \left (a-\frac{b c}{d}\right )+b c \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 (d \cosh \left (a-\frac{b c}{d}\right )+b c \sinh \left (a-\frac{b c}{d}\right )\right )+4 d^3 \sinh (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 \sinh \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*Sinh[a + b*x]*SinhIntegral[c + d*x],x]

[Out]

(2*b^2*d*Cosh[a - c + b*x - d*x] + 2*b*d^2*Cosh[a - c + b*x - d*x] - 2*b^2*d*Cosh[a + c + (b + d)*x] + 2*b*d^2
*Cosh[a + c + (b + d)*x] - 2*(b^2 - d^2)*CoshIntegral[-(((b - d)*(c + d*x))/d)]*(d*Cosh[a - (b*c)/d] + b*c*Sin
h[a - (b*c)/d]) + 2*(b^2 - d^2)*CoshIntegral[((b + d)*(c + d*x))/d]*(d*Cosh[a - (b*c)/d] + b*c*Sinh[a - (b*c)/
d]) + 4*b^3*d*x*Cosh[a + b*x]*SinhIntegral[c + d*x] - 4*b*d^3*x*Cosh[a + b*x]*SinhIntegral[c + d*x] - 4*b^2*d*
Sinh[a + b*x]*SinhIntegral[c + d*x] + 4*d^3*Sinh[a + b*x]*SinhIntegral[c + d*x] - b^3*c*Cosh[a - (b*c)/d]*Sinh
Integral[((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]*SinhIntegr
al[((b - d)*(c + d*x))/d] + 2*b^3*c*Cosh[a - (b*c)/d]*SinhIntegral[((b + d)*(c + d*x))/d] - 2*b*c*d^2*Cosh[a -
(b*c)/d]*SinhIntegral[((b + d)*(c + d*x))/d] + 2*b^2*d*Sinh[a - (b*c)/d]*SinhIntegral[((b + d)*(c + d*x))/d]
- 2*d^3*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]*Si
nhIntegral[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*Sinh
[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 = 0.302, size = 0, normalized size = 0. \begin{align*} \int x{\it Shi} \left ( dx+c \right ) \sinh \left ( bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*Shi(d*x+c)*sinh(b*x+a),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 ) \sinh \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integral(x*sinh(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 \sinh{\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*Shi(d*x+c)*sinh(b*x+a),x)

[Out]

Integral(x*sinh(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 ) \sinh \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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