\(\int x \sinh (a+b x) \text {Shi}(c+d x) \, dx\) [63]

Optimal result

Integrand size = 14, antiderivative size = 371 \[ \int x \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {\cosh (a-c+(b-d) x)}{2 b (b-d)}-\frac {\cosh (a+c+(b+d) x)}{2 b (b+d)}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}-\frac {c \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b d}+\frac {c \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b d}-\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {x \cosh (a+b x) \text {Shi}(c+d x)}{b}-\frac {\sinh (a+b x) \text {Shi}(c+d x)}{b^2}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6677, 6874, 5737, 2718, 5580, 3384, 3379, 3382, 6681, 5578} \[ \int x \sinh (a+b x) \text {Shi}(c+d x) \, dx=-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}-\frac {\sinh (a+b x) \text {Shi}(c+d x)}{b^2}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}-\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {x \cosh (a+b x) \text {Shi}(c+d x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}+\frac {\cosh (a+x (b-d)-c)}{2 b (b-d)}-\frac {\cosh (a+x (b+d)+c)}{2 b (b+d)} \]

[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 2718

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

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 5580

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 5737

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 6677

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 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]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \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 [A] (verified)

Time = 2.61 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.73 \[ \int x \sinh (a+b x) \text {Shi}(c+d x) \, dx=\frac {\frac {e^{-a} \left (b d e^{-c} \left (-\frac {e^{-((b+d) x)}}{b+d}+\frac {e^{2 a+b x-d x}}{b-d}\right )-(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )-(b c-d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )}{d}-\frac {e^{-a} \left (b d e^c \left (\frac {e^{(-b+d) x}}{-b+d}+\frac {e^{2 a+(b+d) x}}{b+d}\right )+(-b c+d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )-(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+4 (b x \cosh (a+b x)-\sinh (a+b x)) \text {Shi}(c+d x)}{4 b^2} \]

[In]

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

[Out]

(((b*d*(-(1/((b + d)*E^((b + d)*x))) + E^(2*a + b*x - d*x)/(b - d)))/E^c - (b*c + d)*E^(2*a - (b*c)/d)*ExpInte
gralEi[((b - d)*(c + d*x))/d] - (b*c - d)*E^((b*c)/d)*ExpIntegralEi[-(((b + d)*(c + d*x))/d)])/(d*E^a) - (b*d*
E^c*(E^((-b + d)*x)/(-b + d) + E^(2*a + (b + d)*x)/(b + d)) + (-(b*c) + d)*E^((b*c)/d)*ExpIntegralEi[-(((b - d
)*(c + d*x))/d)] - (b*c + d)*E^(2*a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(d*E^a) + 4*(b*x*Cosh[a +
b*x] - Sinh[a + b*x])*SinhIntegral[c + d*x])/(4*b^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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