\(\int \text {Chi}(c+d x) \sinh (a+b x) \, dx\) [132]

Optimal result

Integrand size = 13, antiderivative size = 153 \[ \int \text {Chi}(c+d x) \sinh (a+b x) \, dx=-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6682, 5579, 3384, 3379, 3382} \[ \int \text {Chi}(c+d x) \sinh (a+b x) \, dx=-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}+\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b} \]

[In]

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

[Out]

-1/2*(Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/b + (Cosh[a + b*x]*CoshIntegral[c + d*x])/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*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)

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 5579

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*Cosh[(c_.) + (d_.)*(x_)]^(q_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Int
[ExpandTrigReduce[(e + f*x)^m, Cosh[a + b*x]^p*Cosh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ
[p, 0] && IGtQ[q, 0] && IntegerQ[m]

Rule 6682

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

Rubi steps \begin{align*} \text {integral}& = \frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {d \int \frac {\cosh (a+b x) \cosh (c+d x)}{c+d x} \, dx}{b} \\ & = \frac {\cosh (a+b x) \text {Chi}(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 {\cosh (a+b x) \text {Chi}(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 {\cosh (a+b x) \text {Chi}(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 (a+b x) \text {Chi}(c+d x)}{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 \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] (verified)

Time = 0.66 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70 \[ \int \text {Chi}(c+d x) \sinh (a+b x) \, dx=-\frac {-4 \cosh (a+b x) \text {Chi}(c+d x)+e^{-a+\frac {b c}{d}} \left (\operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+\operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )+e^{a-\frac {b c}{d}} \left (\operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )+\operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{4 b} \]

[In]

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

[Out]

-1/4*(-4*Cosh[a + b*x]*CoshIntegral[c + d*x] + E^(-a + (b*c)/d)*(ExpIntegralEi[-(((b - d)*(c + d*x))/d)] + Exp
IntegralEi[-(((b + d)*(c + d*x))/d)]) + E^(a - (b*c)/d)*(ExpIntegralEi[((b - d)*(c + d*x))/d] + ExpIntegralEi[
((b + d)*(c + d*x))/d]))/b

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(coshint(c + d*x)*sinh(a + b*x),x)

[Out]

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