\(\int \frac {\text {Shi}(a+b x)}{x^2} \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 46 \[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\frac {b \text {Chi}(b x) \sinh (a)}{a}+\frac {b \cosh (a) \text {Shi}(b x)}{a}-\frac {b \text {Shi}(a+b x)}{a}-\frac {\text {Shi}(a+b x)}{x} \]

[Out]

b*cosh(a)*Shi(b*x)/a-b*Shi(b*x+a)/a-Shi(b*x+a)/x+b*Chi(b*x)*sinh(a)/a

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6667, 6874, 3384, 3379, 3382} \[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\frac {b \sinh (a) \text {Chi}(b x)}{a}-\frac {b \text {Shi}(a+b x)}{a}-\frac {\text {Shi}(a+b x)}{x}+\frac {b \cosh (a) \text {Shi}(b x)}{a} \]

[In]

Int[SinhIntegral[a + b*x]/x^2,x]

[Out]

(b*CoshIntegral[b*x]*Sinh[a])/a + (b*Cosh[a]*SinhIntegral[b*x])/a - (b*SinhIntegral[a + b*x])/a - SinhIntegral
[a + b*x]/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 6667

Int[((c_.) + (d_.)*(x_))^(m_.)*SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(SinhInte
gral[a + b*x]/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(Sinh[a + b*x]/(a + b*x)), x], x] /
; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rule 6874

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\frac {b x \text {Chi}(b x) \sinh (a)+b x \cosh (a) \text {Shi}(b x)-(a+b x) \text {Shi}(a+b x)}{a x} \]

[In]

Integrate[SinhIntegral[a + b*x]/x^2,x]

[Out]

(b*x*CoshIntegral[b*x]*Sinh[a] + b*x*Cosh[a]*SinhIntegral[b*x] - (a + b*x)*SinhIntegral[a + b*x])/(a*x)

Maple [F]

\[\int \frac {\operatorname {Shi}\left (b x +a \right )}{x^{2}}d x\]

[In]

int(Shi(b*x+a)/x^2,x)

[Out]

int(Shi(b*x+a)/x^2,x)

Fricas [F]

\[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

integrate(Shi(b*x+a)/x^2,x, algorithm="fricas")

[Out]

integral(sinh_integral(b*x + a)/x^2, x)

Sympy [F]

\[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\int \frac {\operatorname {Shi}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

integrate(Shi(b*x+a)/x**2,x)

[Out]

Integral(Shi(a + b*x)/x**2, x)

Maxima [F]

\[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

integrate(Shi(b*x+a)/x^2,x, algorithm="maxima")

[Out]

integrate(Shi(b*x + a)/x^2, x)

Giac [F]

\[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\int { \frac {{\rm Shi}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

integrate(Shi(b*x+a)/x^2,x, algorithm="giac")

[Out]

integrate(Shi(b*x + a)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Shi}(a+b x)}{x^2} \, dx=\int \frac {\mathrm {sinhint}\left (a+b\,x\right )}{x^2} \,d x \]

[In]

int(sinhint(a + b*x)/x^2,x)

[Out]

int(sinhint(a + b*x)/x^2, x)