\(\int x^2 \sinh (b x) \text {Shi}(b x) \, dx\) [44]

Optimal result

Integrand size = 12, antiderivative size = 90 \[ \int x^2 \sinh (b x) \text {Shi}(b x) \, dx=-\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {\text {Shi}(2 b x)}{b^3} \]

[Out]

-5/4*x/b^2+2*cosh(b*x)*Shi(b*x)/b^3+x^2*cosh(b*x)*Shi(b*x)/b-Shi(2*b*x)/b^3+5/4*cosh(b*x)*sinh(b*x)/b^3-2*x*Sh
i(b*x)*sinh(b*x)/b^2-1/2*x*sinh(b*x)^2/b^2

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6677, 12, 5480, 2715, 8, 6683, 6675, 5556, 3379} \[ \int x^2 \sinh (b x) \text {Shi}(b x) \, dx=-\frac {\text {Shi}(2 b x)}{b^3}+\frac {2 \text {Shi}(b x) \cosh (b x)}{b^3}+\frac {5 \sinh (b x) \cosh (b x)}{4 b^3}-\frac {2 x \text {Shi}(b x) \sinh (b x)}{b^2}-\frac {5 x}{4 b^2}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b} \]

[In]

Int[x^2*Sinh[b*x]*SinhIntegral[b*x],x]

[Out]

(-5*x)/(4*b^2) + (5*Cosh[b*x]*Sinh[b*x])/(4*b^3) - (x*Sinh[b*x]^2)/(2*b^2) + (2*Cosh[b*x]*SinhIntegral[b*x])/b
^3 + (x^2*Cosh[b*x]*SinhIntegral[b*x])/b - (2*x*Sinh[b*x]*SinhIntegral[b*x])/b^2 - SinhIntegral[2*b*x]/b^3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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 5480

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[x^(m - n
+ 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

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]

Rule 6677

Int[((e_.) + (f_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e
+ f*x)^m*Cosh[a + b*x]*(SinhIntegral[c + d*x]/b), x] + (-Dist[d/b, Int[(e + f*x)^m*Cosh[a + b*x]*(Sinh[c + d*
x]/(c + d*x)), x], x] - Dist[f*(m/b), Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x], x]) /; Fr
eeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]

Rule 6683

Int[Cosh[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e
+ f*x)^m*Sinh[a + b*x]*(SinhIntegral[c + d*x]/b), x] + (-Dist[d/b, Int[(e + f*x)^m*Sinh[a + b*x]*(Sinh[c + d*
x]/(c + d*x)), x], x] - Dist[f*(m/b), Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*SinhIntegral[c + d*x], x], x]) /; Fr
eeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 \int x \cosh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {x \cosh (b x) \sinh (b x)}{b} \, dx \\ & = \frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \int \sinh (b x) \text {Shi}(b x) \, dx}{b^2}-\frac {\int x \cosh (b x) \sinh (b x) \, dx}{b}+\frac {2 \int \frac {\sinh ^2(b x)}{b} \, dx}{b} \\ & = -\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {\int \sinh ^2(b x) \, dx}{2 b^2}-\frac {2 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b^2}+\frac {2 \int \sinh ^2(b x) \, dx}{b^2} \\ & = \frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {2 \int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b^3}-\frac {\int 1 \, dx}{4 b^2}-\frac {\int 1 \, dx}{b^2} \\ & = -\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {2 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b^3} \\ & = -\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{b^3} \\ & = -\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {\text {Shi}(2 b x)}{b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.71 \[ \int x^2 \sinh (b x) \text {Shi}(b x) \, dx=\frac {-8 b x-2 b x \cosh (2 b x)+5 \sinh (2 b x)+8 \left (\left (2+b^2 x^2\right ) \cosh (b x)-2 b x \sinh (b x)\right ) \text {Shi}(b x)-8 \text {Shi}(2 b x)}{8 b^3} \]

[In]

Integrate[x^2*Sinh[b*x]*SinhIntegral[b*x],x]

[Out]

(-8*b*x - 2*b*x*Cosh[2*b*x] + 5*Sinh[2*b*x] + 8*((2 + b^2*x^2)*Cosh[b*x] - 2*b*x*Sinh[b*x])*SinhIntegral[b*x]
- 8*SinhIntegral[2*b*x])/(8*b^3)

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right )^{2}}{2}+\frac {5 \cosh \left (b x \right ) \sinh \left (b x \right )}{4}-\frac {3 b x}{4}-\operatorname {Shi}\left (2 b x \right )}{b^{3}}\)	\(68\)
default	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right )^{2}}{2}+\frac {5 \cosh \left (b x \right ) \sinh \left (b x \right )}{4}-\frac {3 b x}{4}-\operatorname {Shi}\left (2 b x \right )}{b^{3}}\)	\(68\)

[In]

int(x^2*Shi(b*x)*sinh(b*x),x,method=_RETURNVERBOSE)

[Out]

1/b^3*(Shi(b*x)*(b^2*x^2*cosh(b*x)-2*b*x*sinh(b*x)+2*cosh(b*x))-1/2*b*x*cosh(b*x)^2+5/4*cosh(b*x)*sinh(b*x)-3/
4*b*x-Shi(2*b*x))

Fricas [F]

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

[In]

integrate(x^2*Shi(b*x)*sinh(b*x),x, algorithm="fricas")

[Out]

integral(x^2*sinh(b*x)*sinh_integral(b*x), x)

Sympy [F]

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

[In]

integrate(x**2*Shi(b*x)*sinh(b*x),x)

[Out]

Integral(x**2*sinh(b*x)*Shi(b*x), x)

Maxima [F]

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

[In]

integrate(x^2*Shi(b*x)*sinh(b*x),x, algorithm="maxima")

[Out]

integrate(x^2*Shi(b*x)*sinh(b*x), x)

Giac [F]

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

[In]

integrate(x^2*Shi(b*x)*sinh(b*x),x, algorithm="giac")

[Out]

integrate(x^2*Shi(b*x)*sinh(b*x), x)

Mupad [F(-1)]

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

[In]

int(x^2*sinhint(b*x)*sinh(b*x),x)

[Out]

int(x^2*sinhint(b*x)*sinh(b*x), x)