\(\int x^2 \cosh (b x) \text {Chi}(b x) \, dx\) [112]

Optimal result

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

[Out]

3/4*x/b^2-2*x*Chi(b*x)*cosh(b*x)/b^2-Shi(2*b*x)/b^3+2*Chi(b*x)*sinh(b*x)/b^3+x^2*Chi(b*x)*sinh(b*x)/b+5/4*cosh
(b*x)*sinh(b*x)/b^3-1/2*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 = {6678, 12, 5480, 2715, 8, 6684, 6676, 5556, 3379} \[ \int x^2 \cosh (b x) \text {Chi}(b x) \, dx=\frac {2 \text {Chi}(b x) \sinh (b x)}{b^3}-\frac {\text {Shi}(2 b x)}{b^3}+\frac {5 \sinh (b x) \cosh (b x)}{4 b^3}-\frac {2 x \text {Chi}(b x) \cosh (b x)}{b^2}+\frac {3 x}{4 b^2}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {x^2 \text {Chi}(b x) \sinh (b x)}{b} \]

[In]

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

[Out]

(3*x)/(4*b^2) - (2*x*Cosh[b*x]*CoshIntegral[b*x])/b^2 + (5*Cosh[b*x]*Sinh[b*x])/(4*b^3) + (2*CoshIntegral[b*x]
*Sinh[b*x])/b^3 + (x^2*CoshIntegral[b*x]*Sinh[b*x])/b - (x*Sinh[b*x]^2)/(2*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 6676

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

Rule 6678

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

Rule 6684

Int[CoshIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(e
+ f*x)^m*Cosh[a + b*x]*(CoshIntegral[c + d*x]/b), x] + (-Dist[d/b, Int[(e + f*x)^m*Cosh[a + b*x]*(Cosh[c + d*
x]/(c + d*x)), x], x] - Dist[f*(m/b), Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*CoshIntegral[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 \text {Chi}(b x) \sinh (b x)}{b}-\frac {2 \int x \text {Chi}(b x) \sinh (b x) \, dx}{b}-\int \frac {x \cosh (b x) \sinh (b x)}{b} \, dx \\ & = -\frac {2 x \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {x^2 \text {Chi}(b x) \sinh (b x)}{b}+\frac {2 \int \cosh (b x) \text {Chi}(b x) \, dx}{b^2}-\frac {\int x \cosh (b x) \sinh (b x) \, dx}{b}+\frac {2 \int \frac {\cosh ^2(b x)}{b} \, dx}{b} \\ & = -\frac {2 x \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {2 \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^2 \text {Chi}(b x) \sinh (b x)}{b}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {\int \sinh ^2(b x) \, dx}{2 b^2}+\frac {2 \int \cosh ^2(b x) \, dx}{b^2}-\frac {2 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b^2} \\ & = -\frac {2 x \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}+\frac {2 \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^2 \text {Chi}(b x) \sinh (b x)}{b}-\frac {x \sinh ^2(b x)}{2 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 {3 x}{4 b^2}-\frac {2 x \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}+\frac {2 \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^2 \text {Chi}(b x) \sinh (b x)}{b}-\frac {x \sinh ^2(b x)}{2 b^2}-\frac {2 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b^3} \\ & = \frac {3 x}{4 b^2}-\frac {2 x \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}+\frac {2 \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^2 \text {Chi}(b x) \sinh (b x)}{b}-\frac {x \sinh ^2(b x)}{2 b^2}-\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{b^3} \\ & = \frac {3 x}{4 b^2}-\frac {2 x \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}+\frac {2 \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^2 \text {Chi}(b x) \sinh (b x)}{b}-\frac {x \sinh ^2(b x)}{2 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 \[ \int x^2 \cosh (b x) \text {Chi}(b x) \, dx=\frac {8 b x-2 b x \cosh (2 b x)+8 \text {Chi}(b x) \left (-2 b x \cosh (b x)+\left (2+b^2 x^2\right ) \sinh (b x)\right )+5 \sinh (2 b x)-8 \text {Shi}(2 b x)}{8 b^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*cosh(b*x)*Chi(b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*Chi(b*x)*cosh(b*x), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*Chi(b*x)*cosh(b*x), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*coshint(b*x)*cosh(b*x), x)