\(\int \sinh (a+b x) \text {Shi}(a+b x) \, dx\) [57]

Optimal result

Integrand size = 13, antiderivative size = 33 \[ \int \sinh (a+b x) \text {Shi}(a+b x) \, dx=\frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b} \]

[Out]

cosh(b*x+a)*Shi(b*x+a)/b-1/2*Shi(2*b*x+2*a)/b

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6675, 5556, 12, 3379} \[ \int \sinh (a+b x) \text {Shi}(a+b x) \, dx=\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b} \]

[In]

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

[Out]

(Cosh[a + b*x]*SinhIntegral[a + b*x])/b - SinhIntegral[2*a + 2*b*x]/(2*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 5556

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

Rule 6675

Int[Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cosh[a + b*x]*(SinhIntegral[c
+ d*x]/b), x] - Dist[d/b, Int[Cosh[a + b*x]*(Sinh[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 {Shi}(a+b x)}{b}-\int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx \\ & = \frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx \\ & = \frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx \\ & = \frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \sinh (a+b x) \text {Shi}(a+b x) \, dx=\frac {\cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {\text {Shi}(2 (a+b x))}{2 b} \]

[In]

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

[Out]

(Cosh[a + b*x]*SinhIntegral[a + b*x])/b - SinhIntegral[2*(a + b*x)]/(2*b)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\cosh \left (b x +a \right ) \operatorname {Shi}\left (b x +a \right )-\frac {\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b}\)	\(30\)
default	\(\frac {\cosh \left (b x +a \right ) \operatorname {Shi}\left (b x +a \right )-\frac {\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b}\)	\(30\)

[In]

int(Shi(b*x+a)*sinh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b*(cosh(b*x+a)*Shi(b*x+a)-1/2*Shi(2*b*x+2*a))

Fricas [F]

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

[In]

integrate(Shi(b*x+a)*sinh(b*x+a),x, algorithm="fricas")

[Out]

integral(sinh(b*x + a)*sinh_integral(b*x + a), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sinhint(a + b*x)*sinh(a + b*x),x)

[Out]

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