\(\int x^2 \cosh (b x) \text {Shi}(b x) \, dx\) [51]

Optimal result

Integrand size = 12, antiderivative size = 98 \[ \int x^2 \cosh (b x) \text {Shi}(b x) \, dx=\frac {x^2}{4 b}-\frac {\text {Chi}(2 b x)}{b^3}+\frac {\log (x)}{b^3}-\frac {x \cosh (b x) \sinh (b x)}{2 b^2}+\frac {5 \sinh ^2(b x)}{4 b^3}-\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6683, 12, 3391, 30, 6677, 2644, 6681, 3393, 3382} \[ \int x^2 \cosh (b x) \text {Shi}(b x) \, dx=-\frac {\text {Chi}(2 b x)}{b^3}+\frac {2 \text {Shi}(b x) \sinh (b x)}{b^3}+\frac {\log (x)}{b^3}+\frac {5 \sinh ^2(b x)}{4 b^3}-\frac {2 x \text {Shi}(b x) \cosh (b x)}{b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b^2}+\frac {x^2 \text {Shi}(b x) \sinh (b x)}{b}+\frac {x^2}{4 b} \]

[In]

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

[Out]

x^2/(4*b) - CoshIntegral[2*b*x]/b^3 + Log[x]/b^3 - (x*Cosh[b*x]*Sinh[b*x])/(2*b^2) + (5*Sinh[b*x]^2)/(4*b^3) -
(2*x*Cosh[b*x]*SinhIntegral[b*x])/b^2 + (2*Sinh[b*x]*SinhIntegral[b*x])/b^3 + (x^2*Sinh[b*x]*SinhIntegral[b*x
])/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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

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 3391

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

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 6681

Int[Cosh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sinh[a + b*x]*(SinhIntegral[c
+ d*x]/b), x] - Dist[d/b, Int[Sinh[a + b*x]*(Sinh[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]

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 \sinh (b x) \text {Shi}(b x)}{b}-\frac {2 \int x \sinh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {x \sinh ^2(b x)}{b} \, dx \\ & = -\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b}+\frac {2 \int \cosh (b x) \text {Shi}(b x) \, dx}{b^2}-\frac {\int x \sinh ^2(b x) \, dx}{b}+\frac {2 \int \frac {\cosh (b x) \sinh (b x)}{b} \, dx}{b} \\ & = -\frac {x \cosh (b x) \sinh (b x)}{2 b^2}+\frac {\sinh ^2(b x)}{4 b^3}-\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b}+\frac {2 \int \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac {2 \int \frac {\sinh ^2(b x)}{b x} \, dx}{b^2}+\frac {\int x \, dx}{2 b} \\ & = \frac {x^2}{4 b}-\frac {x \cosh (b x) \sinh (b x)}{2 b^2}+\frac {\sinh ^2(b x)}{4 b^3}-\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b}-\frac {2 \int \frac {\sinh ^2(b x)}{x} \, dx}{b^3}-\frac {2 \text {Subst}(\int x \, dx,x,i \sinh (b x))}{b^3} \\ & = \frac {x^2}{4 b}-\frac {x \cosh (b x) \sinh (b x)}{2 b^2}+\frac {5 \sinh ^2(b x)}{4 b^3}-\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b}+\frac {2 \int \left (\frac {1}{2 x}-\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b^3} \\ & = \frac {x^2}{4 b}+\frac {\log (x)}{b^3}-\frac {x \cosh (b x) \sinh (b x)}{2 b^2}+\frac {5 \sinh ^2(b x)}{4 b^3}-\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b}-\frac {\int \frac {\cosh (2 b x)}{x} \, dx}{b^3} \\ & = \frac {x^2}{4 b}-\frac {\text {Chi}(2 b x)}{b^3}+\frac {\log (x)}{b^3}-\frac {x \cosh (b x) \sinh (b x)}{2 b^2}+\frac {5 \sinh ^2(b x)}{4 b^3}-\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.73 \[ \int x^2 \cosh (b x) \text {Shi}(b x) \, dx=\frac {2 b^2 x^2+5 \cosh (2 b x)-8 \text {Chi}(2 b x)+8 \log (x)-2 b x \sinh (2 b x)+8 \left (-2 b x \cosh (b x)+\left (2+b^2 x^2\right ) \sinh (b x)\right ) \text {Shi}(b x)}{8 b^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \sinh \left (b x \right )-2 b x \cosh \left (b x \right )+2 \sinh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{2} x^{2}}{4}+\frac {5 \cosh \left (b x \right )^{2}}{4}+\ln \left (b x \right )-\operatorname {Chi}\left (2 b x \right )}{b^{3}}\)	\(76\)
default	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \sinh \left (b x \right )-2 b x \cosh \left (b x \right )+2 \sinh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{2} x^{2}}{4}+\frac {5 \cosh \left (b x \right )^{2}}{4}+\ln \left (b x \right )-\operatorname {Chi}\left (2 b x \right )}{b^{3}}\)	\(76\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*cosh(b*x)*Shi(b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*Shi(b*x)*cosh(b*x), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*Shi(b*x)*cosh(b*x), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*sinhint(b*x)*cosh(b*x), x)