3.79 \(\int x^2 \text{Chi}(b x)^2 \, dx\)

Optimal. Leaf size=112 \[ -\frac{4 \text{Chi}(b x) \sinh (b x)}{3 b^3}+\frac{4 x \text{Chi}(b x) \cosh (b x)}{3 b^2}+\frac{2 \text{Shi}(2 b x)}{3 b^3}-\frac{x}{2 b^2}+\frac{x \sinh ^2(b x)}{3 b^2}-\frac{5 \sinh (b x) \cosh (b x)}{6 b^3}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b} \]

[Out]

-x/(2*b^2) + (4*x*Cosh[b*x]*CoshIntegral[b*x])/(3*b^2) + (x^3*CoshIntegral[b*x]^2)/3 - (5*Cosh[b*x]*Sinh[b*x])
/(6*b^3) - (4*CoshIntegral[b*x]*Sinh[b*x])/(3*b^3) - (2*x^2*CoshIntegral[b*x]*Sinh[b*x])/(3*b) + (x*Sinh[b*x]^
2)/(3*b^2) + (2*SinhIntegral[2*b*x])/(3*b^3)

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Rubi [A]  time = 0.14682, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6537, 6543, 12, 5372, 2635, 8, 6549, 6541, 5448, 3298} \[ -\frac{4 \text{Chi}(b x) \sinh (b x)}{3 b^3}+\frac{4 x \text{Chi}(b x) \cosh (b x)}{3 b^2}+\frac{2 \text{Shi}(2 b x)}{3 b^3}-\frac{x}{2 b^2}+\frac{x \sinh ^2(b x)}{3 b^2}-\frac{5 \sinh (b x) \cosh (b x)}{6 b^3}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(2*b^2) + (4*x*Cosh[b*x]*CoshIntegral[b*x])/(3*b^2) + (x^3*CoshIntegral[b*x]^2)/3 - (5*Cosh[b*x]*Sinh[b*x])
/(6*b^3) - (4*CoshIntegral[b*x]*Sinh[b*x])/(3*b^3) - (2*x^2*CoshIntegral[b*x]*Sinh[b*x])/(3*b) + (x*Sinh[b*x]^
2)/(3*b^2) + (2*SinhIntegral[2*b*x])/(3*b^3)

Rule 6537

Int[CoshIntegral[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*CoshIntegral[b*x]^2)/(m + 1), x] - Dis
t[2/(m + 1), Int[x^m*Cosh[b*x]*CoshIntegral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]

Rule 6543

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 12

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

Rule 5372

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 6549

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]

Rule 6541

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 5448

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 3298

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]

Rubi steps

\begin{align*} \int x^2 \text{Chi}(b x)^2 \, dx &=\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{2}{3} \int x^2 \cosh (b x) \text{Chi}(b x) \, dx\\ &=\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b}+\frac{2}{3} \int \frac{x \cosh (b x) \sinh (b x)}{b} \, dx+\frac{4 \int x \text{Chi}(b x) \sinh (b x) \, dx}{3 b}\\ &=\frac{4 x \cosh (b x) \text{Chi}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b}-\frac{4 \int \cosh (b x) \text{Chi}(b x) \, dx}{3 b^2}+\frac{2 \int x \cosh (b x) \sinh (b x) \, dx}{3 b}-\frac{4 \int \frac{\cosh ^2(b x)}{b} \, dx}{3 b}\\ &=\frac{4 x \cosh (b x) \text{Chi}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{4 \text{Chi}(b x) \sinh (b x)}{3 b^3}-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b}+\frac{x \sinh ^2(b x)}{3 b^2}-\frac{\int \sinh ^2(b x) \, dx}{3 b^2}-\frac{4 \int \cosh ^2(b x) \, dx}{3 b^2}+\frac{4 \int \frac{\cosh (b x) \sinh (b x)}{b x} \, dx}{3 b^2}\\ &=\frac{4 x \cosh (b x) \text{Chi}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{5 \cosh (b x) \sinh (b x)}{6 b^3}-\frac{4 \text{Chi}(b x) \sinh (b x)}{3 b^3}-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b}+\frac{x \sinh ^2(b x)}{3 b^2}+\frac{4 \int \frac{\cosh (b x) \sinh (b x)}{x} \, dx}{3 b^3}+\frac{\int 1 \, dx}{6 b^2}-\frac{2 \int 1 \, dx}{3 b^2}\\ &=-\frac{x}{2 b^2}+\frac{4 x \cosh (b x) \text{Chi}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{5 \cosh (b x) \sinh (b x)}{6 b^3}-\frac{4 \text{Chi}(b x) \sinh (b x)}{3 b^3}-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b}+\frac{x \sinh ^2(b x)}{3 b^2}+\frac{4 \int \frac{\sinh (2 b x)}{2 x} \, dx}{3 b^3}\\ &=-\frac{x}{2 b^2}+\frac{4 x \cosh (b x) \text{Chi}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{5 \cosh (b x) \sinh (b x)}{6 b^3}-\frac{4 \text{Chi}(b x) \sinh (b x)}{3 b^3}-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b}+\frac{x \sinh ^2(b x)}{3 b^2}+\frac{2 \int \frac{\sinh (2 b x)}{x} \, dx}{3 b^3}\\ &=-\frac{x}{2 b^2}+\frac{4 x \cosh (b x) \text{Chi}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(b x)^2-\frac{5 \cosh (b x) \sinh (b x)}{6 b^3}-\frac{4 \text{Chi}(b x) \sinh (b x)}{3 b^3}-\frac{2 x^2 \text{Chi}(b x) \sinh (b x)}{3 b}+\frac{x \sinh ^2(b x)}{3 b^2}+\frac{2 \text{Shi}(2 b x)}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0725922, size = 78, normalized size = 0.7 \[ \frac{4 b^3 x^3 \text{Chi}(b x)^2-8 \text{Chi}(b x) \left (\left (b^2 x^2+2\right ) \sinh (b x)-2 b x \cosh (b x)\right )+8 \text{Shi}(2 b x)-8 b x-5 \sinh (2 b x)+2 b x \cosh (2 b x)}{12 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*x + 2*b*x*Cosh[2*b*x] + 4*b^3*x^3*CoshIntegral[b*x]^2 - 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])/(12*b^3)

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Maple [A]  time = 0.056, size = 84, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3} \left ({\it Chi} \left ( bx \right ) \right ) ^{2}}{3}}-2\,{\it Chi} \left ( bx \right ) \left ( 1/3\,{b}^{2}{x}^{2}\sinh \left ( bx \right ) -2/3\,bx\cosh \left ( bx \right ) +2/3\,\sinh \left ( bx \right ) \right ) +{\frac{bx \left ( \cosh \left ( bx \right ) \right ) ^{2}}{3}}-{\frac{5\,\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{6}}-{\frac{5\,bx}{6}}+{\frac{2\,{\it Shi} \left ( 2\,bx \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*Chi(b*x)^2,x)

[Out]

1/b^3*(1/3*b^3*x^3*Chi(b*x)^2-2*Chi(b*x)*(1/3*b^2*x^2*sinh(b*x)-2/3*b*x*cosh(b*x)+2/3*sinh(b*x))+1/3*b*x*cosh(
b*x)^2-5/6*cosh(b*x)*sinh(b*x)-5/6*b*x+2/3*Shi(2*b*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Chi}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*Chi(b*x)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{Chi}\left (b x\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2*cosh_integral(b*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{Chi}^{2}\left (b x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*Chi(b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Chi}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*Chi(b*x)^2, x)