\(\int x \text {Shi}(a+b x) \, dx\) [20]

Optimal result

Integrand size = 8, antiderivative size = 71 \[ \int x \text {Shi}(a+b x) \, dx=\frac {a \cosh (a+b x)}{2 b^2}-\frac {x \cosh (a+b x)}{2 b}+\frac {\sinh (a+b x)}{2 b^2}-\frac {a^2 \text {Shi}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(a+b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6667, 6874, 2718, 3377, 2717, 3379} \[ \int x \text {Shi}(a+b x) \, dx=-\frac {a^2 \text {Shi}(a+b x)}{2 b^2}+\frac {\sinh (a+b x)}{2 b^2}+\frac {a \cosh (a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(a+b x)-\frac {x \cosh (a+b x)}{2 b} \]

[In]

Int[x*SinhIntegral[a + b*x],x]

[Out]

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 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 {1}{2} x^2 \text {Shi}(a+b x)-\frac {1}{2} b \int \frac {x^2 \sinh (a+b x)}{a+b x} \, dx \\ & = \frac {1}{2} x^2 \text {Shi}(a+b x)-\frac {1}{2} b \int \left (-\frac {a \sinh (a+b x)}{b^2}+\frac {x \sinh (a+b x)}{b}+\frac {a^2 \sinh (a+b x)}{b^2 (a+b x)}\right ) \, dx \\ & = \frac {1}{2} x^2 \text {Shi}(a+b x)-\frac {1}{2} \int x \sinh (a+b x) \, dx+\frac {a \int \sinh (a+b x) \, dx}{2 b}-\frac {a^2 \int \frac {\sinh (a+b x)}{a+b x} \, dx}{2 b} \\ & = \frac {a \cosh (a+b x)}{2 b^2}-\frac {x \cosh (a+b x)}{2 b}-\frac {a^2 \text {Shi}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(a+b x)+\frac {\int \cosh (a+b x) \, dx}{2 b} \\ & = \frac {a \cosh (a+b x)}{2 b^2}-\frac {x \cosh (a+b x)}{2 b}+\frac {\sinh (a+b x)}{2 b^2}-\frac {a^2 \text {Shi}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(a+b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.66 \[ \int x \text {Shi}(a+b x) \, dx=\frac {(a-b x) \cosh (a+b x)+\sinh (a+b x)+\left (-a^2+b^2 x^2\right ) \text {Shi}(a+b x)}{2 b^2} \]

[In]

Integrate[x*SinhIntegral[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82

method	result	size
parts	\(\frac {x^{2} \operatorname {Shi}\left (b x +a \right )}{2}-\frac {a^{2} \operatorname {Shi}\left (b x +a \right )-2 a \cosh \left (b x +a \right )+\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )}{2 b^{2}}\)	\(58\)
derivativedivides	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )+a \cosh \left (b x +a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )}{2}+\frac {\sinh \left (b x +a \right )}{2}}{b^{2}}\)	\(60\)
default	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )+a \cosh \left (b x +a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )}{2}+\frac {\sinh \left (b x +a \right )}{2}}{b^{2}}\)	\(60\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x*sinh_integral(b*x + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*Shi(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x*Shi(b*x + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(x*Shi(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int x \text {Shi}(a+b x) \, dx=\frac {\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a+{\mathrm {e}}^{2\,a+2\,b\,x}+a\,{\mathrm {e}}^{2\,a+2\,b\,x}-2\,a^2\,\mathrm {sinhint}\left (a+b\,x\right )\,{\mathrm {e}}^{a+b\,x}-1\right )}{4}-\frac {b\,{\mathrm {e}}^{-a-b\,x}\,\left (x+x\,{\mathrm {e}}^{2\,a+2\,b\,x}\right )}{4}}{b^2}+\frac {x^2\,\mathrm {sinhint}\left (a+b\,x\right )}{2} \]

[In]

int(x*sinhint(a + b*x),x)

[Out]

((exp(- a - b*x)*(a + exp(2*a + 2*b*x) + a*exp(2*a + 2*b*x) - 2*a^2*sinhint(a + b*x)*exp(a + b*x) - 1))/4 - (b
*exp(- a - b*x)*(x + x*exp(2*a + 2*b*x)))/4)/b^2 + (x^2*sinhint(a + b*x))/2