\(\int x \text {Shi}(b x) \, dx\) [4]

Optimal result

Integrand size = 6, antiderivative size = 35 \[ \int x \text {Shi}(b x) \, dx=-\frac {x \cosh (b x)}{2 b}+\frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6667, 12, 3377, 2717} \[ \int x \text {Shi}(b x) \, dx=\frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x)-\frac {x \cosh (b x)}{2 b} \]

[In]

Int[x*SinhIntegral[b*x],x]

[Out]

-1/2*(x*Cosh[b*x])/b + Sinh[b*x]/(2*b^2) + (x^2*SinhIntegral[b*x])/2

Rule 12

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

Rule 2717

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {Shi}(b x)-\frac {1}{2} b \int \frac {x \sinh (b x)}{b} \, dx \\ & = \frac {1}{2} x^2 \text {Shi}(b x)-\frac {1}{2} \int x \sinh (b x) \, dx \\ & = -\frac {x \cosh (b x)}{2 b}+\frac {1}{2} x^2 \text {Shi}(b x)+\frac {\int \cosh (b x) \, dx}{2 b} \\ & = -\frac {x \cosh (b x)}{2 b}+\frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int x \text {Shi}(b x) \, dx=-\frac {x \cosh (b x)}{2 b}+\frac {\sinh (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Shi}(b x) \]

[In]

Integrate[x*SinhIntegral[b*x],x]

[Out]

-1/2*(x*Cosh[b*x])/b + Sinh[b*x]/(2*b^2) + (x^2*SinhIntegral[b*x])/2

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {x^{2} \operatorname {Shi}\left (b x \right )}{2}-\frac {b x \cosh \left (b x \right )-\sinh \left (b x \right )}{2 b^{2}}\)	\(30\)
derivativedivides	\(\frac {\frac {b^{2} x^{2} \operatorname {Shi}\left (b x \right )}{2}-\frac {b x \cosh \left (b x \right )}{2}+\frac {\sinh \left (b x \right )}{2}}{b^{2}}\)	\(32\)
default	\(\frac {\frac {b^{2} x^{2} \operatorname {Shi}\left (b x \right )}{2}-\frac {b x \cosh \left (b x \right )}{2}+\frac {\sinh \left (b x \right )}{2}}{b^{2}}\)	\(32\)
meijerg	\(\frac {i \sqrt {\pi }\, \left (\frac {i b x \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {i \sinh \left (b x \right )}{2 \sqrt {\pi }}-\frac {i b^{2} x^{2} \operatorname {Shi}\left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}\)	\(49\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x*sinh_integral(b*x), x)

Sympy [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int x \text {Shi}(b x) \, dx=\frac {x^{2} \operatorname {Shi}{\left (b x \right )}}{2} - \frac {x \cosh {\left (b x \right )}}{2 b} + \frac {\sinh {\left (b x \right )}}{2 b^{2}} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(x*Shi(b*x), x)

Giac [F]

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

[In]

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

[Out]

integrate(x*Shi(b*x), x)

Mupad [F(-1)]

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

[In]

int(x*sinhint(b*x),x)

[Out]

(sinh(b*x)/2 - (b*x*cosh(b*x))/2)/b^2 + (x^2*sinhint(b*x))/2