\(\int x \text {Chi}(c+d x) \sinh (a+b x) \, dx\) [131]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6684, 5762, 6874, 2717, 3384, 3379, 3382, 6676, 5580} \[ \int x \text {Chi}(c+d x) \sinh (a+b x) \, dx=\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}-\frac {\sinh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {\sinh (a+x (b-d)-c)}{2 b (b-d)}-\frac {\sinh (a+x (b+d)+c)}{2 b (b+d)} \]

[In]

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

[Out]

(c*Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (x*Cosh[a + b*x]*CoshIntegral[c + d*x]
)/b + (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) + (CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b^2) - (C
oshIntegral[c + d*x]*Sinh[a + b*x])/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*Sinh[a - (b*c)/d]*SinhInt
egral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 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 5762

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

Rule 6676

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

Rule 6684

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

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 {Chi}(c+d x)}{b}-\frac {\int \cosh (a+b x) \text {Chi}(c+d x) \, dx}{b}-\frac {d \int \frac {x \cosh (a+b x) \cosh (c+d x)}{c+d x} \, dx}{b} \\ & = \frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}+\frac {d \int \frac {\cosh (c+d x) \sinh (a+b 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 {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}+\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 {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}+\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 {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}-\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 {x \cosh (a+b x) \text {Chi}(c+d x)}{b}+\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 {\text {Chi}(c+d x) \sinh (a+b x)}{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 \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 {x \cosh (a+b x) \text {Chi}(c+d x)}{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 {\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 {\text {Chi}(c+d x) \sinh (a+b x)}{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 \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 [A] (verified)

Time = 2.40 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.87 \[ \int x \text {Chi}(c+d x) \sinh (a+b x) \, dx=\frac {e^{-a-c-(b+d) x} \left (b d \left (d \left (-1+e^{2 (c+d x)}\right )+b \left (1+e^{2 (c+d x)}\right )\right )+(b c-d) \left (b^2-d^2\right ) e^{\frac {(b+d) (c+d x)}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+(b c-d) \left (b^2-d^2\right ) e^{\frac {(b+d) (c+d x)}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )+e^{a-\frac {c (b+d)}{d}} \left (-b d e^{\frac {b c}{d}+b x-d x} \left (b+d+b e^{2 (c+d x)}-d e^{2 (c+d x)}\right )+(b c+d) \left (b^2-d^2\right ) e^c \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )+(b c+d) \left (b^2-d^2\right ) e^c \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )+4 (b-d) d (b+d) \text {Chi}(c+d x) (b x \cosh (a+b x)-\sinh (a+b x))}{4 b^2 (b-d) d (b+d)} \]

[In]

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

[Out]

(E^(-a - c - (b + d)*x)*(b*d*(d*(-1 + E^(2*(c + d*x))) + b*(1 + E^(2*(c + d*x)))) + (b*c - d)*(b^2 - d^2)*E^((
(b + d)*(c + d*x))/d)*ExpIntegralEi[-(((b - d)*(c + d*x))/d)] + (b*c - d)*(b^2 - d^2)*E^(((b + d)*(c + d*x))/d
)*ExpIntegralEi[-(((b + d)*(c + d*x))/d)]) + E^(a - (c*(b + d))/d)*(-(b*d*E^((b*c)/d + b*x - d*x)*(b + d + b*E
^(2*(c + d*x)) - d*E^(2*(c + d*x)))) + (b*c + d)*(b^2 - d^2)*E^c*ExpIntegralEi[((b - d)*(c + d*x))/d] + (b*c +
d)*(b^2 - d^2)*E^c*ExpIntegralEi[((b + d)*(c + d*x))/d]) + 4*(b - d)*d*(b + d)*CoshIntegral[c + d*x]*(b*x*Cos
h[a + b*x] - Sinh[a + b*x]))/(4*b^2*(b - d)*d*(b + d))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(x*Chi(d*x+c)*sinh(b*x+a),x, algorithm="fricas")

[Out]

integral(x*cosh_integral(d*x + c)*sinh(b*x + a), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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