3.120 \(\int x^3 \text{Chi}(b x) \sinh (b x) \, dx\)

Optimal. Leaf size=146 \[ -\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}-\frac{6 \text{Chi}(b x) \sinh (b x)}{b^4}+\frac{6 x \text{Chi}(b x) \cosh (b x)}{b^3}+\frac{3 \text{Shi}(2 b x)}{b^4}-\frac{x^2 \sinh (b x) \cosh (b x)}{2 b^2}-\frac{5 x}{2 b^3}+\frac{3 x \sinh ^2(b x)}{2 b^3}+\frac{x \cosh ^2(b x)}{2 b^3}-\frac{4 \sinh (b x) \cosh (b x)}{b^4}+\frac{x^3 \text{Chi}(b x) \cosh (b x)}{b}-\frac{x^3}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.188602, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6549, 12, 3311, 30, 2635, 8, 6543, 5372, 6541, 5448, 3298} \[ -\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}-\frac{6 \text{Chi}(b x) \sinh (b x)}{b^4}+\frac{6 x \text{Chi}(b x) \cosh (b x)}{b^3}+\frac{3 \text{Shi}(2 b x)}{b^4}-\frac{x^2 \sinh (b x) \cosh (b x)}{2 b^2}-\frac{5 x}{2 b^3}+\frac{3 x \sinh ^2(b x)}{2 b^3}+\frac{x \cosh ^2(b x)}{2 b^3}-\frac{4 \sinh (b x) \cosh (b x)}{b^4}+\frac{x^3 \text{Chi}(b x) \cosh (b x)}{b}-\frac{x^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*CoshIntegral[b*x]*Sinh[b*x],x]

[Out]

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

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 12

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

Rule 3311

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -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 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 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 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^3 \text{Chi}(b x) \sinh (b x) \, dx &=\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{3 \int x^2 \cosh (b x) \text{Chi}(b x) \, dx}{b}-\int \frac{x^2 \cosh ^2(b x)}{b} \, dx\\ &=\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{6 \int x \text{Chi}(b x) \sinh (b x) \, dx}{b^2}-\frac{\int x^2 \cosh ^2(b x) \, dx}{b}+\frac{3 \int \frac{x \cosh (b x) \sinh (b x)}{b} \, dx}{b}\\ &=\frac{x \cosh ^2(b x)}{2 b^3}+\frac{6 x \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}-\frac{\int \cosh ^2(b x) \, dx}{2 b^3}-\frac{6 \int \cosh (b x) \text{Chi}(b x) \, dx}{b^3}+\frac{3 \int x \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac{6 \int \frac{\cosh ^2(b x)}{b} \, dx}{b^2}-\frac{\int x^2 \, dx}{2 b}\\ &=-\frac{x^3}{6 b}+\frac{x \cosh ^2(b x)}{2 b^3}+\frac{6 x \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{\cosh (b x) \sinh (b x)}{4 b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac{6 \text{Chi}(b x) \sinh (b x)}{b^4}-\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{3 x \sinh ^2(b x)}{2 b^3}-\frac{\int 1 \, dx}{4 b^3}-\frac{3 \int \sinh ^2(b x) \, dx}{2 b^3}-\frac{6 \int \cosh ^2(b x) \, dx}{b^3}+\frac{6 \int \frac{\cosh (b x) \sinh (b x)}{b x} \, dx}{b^3}\\ &=-\frac{x}{4 b^3}-\frac{x^3}{6 b}+\frac{x \cosh ^2(b x)}{2 b^3}+\frac{6 x \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac{6 \text{Chi}(b x) \sinh (b x)}{b^4}-\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{3 x \sinh ^2(b x)}{2 b^3}+\frac{6 \int \frac{\cosh (b x) \sinh (b x)}{x} \, dx}{b^4}+\frac{3 \int 1 \, dx}{4 b^3}-\frac{3 \int 1 \, dx}{b^3}\\ &=-\frac{5 x}{2 b^3}-\frac{x^3}{6 b}+\frac{x \cosh ^2(b x)}{2 b^3}+\frac{6 x \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac{6 \text{Chi}(b x) \sinh (b x)}{b^4}-\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{3 x \sinh ^2(b x)}{2 b^3}+\frac{6 \int \frac{\sinh (2 b x)}{2 x} \, dx}{b^4}\\ &=-\frac{5 x}{2 b^3}-\frac{x^3}{6 b}+\frac{x \cosh ^2(b x)}{2 b^3}+\frac{6 x \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac{6 \text{Chi}(b x) \sinh (b x)}{b^4}-\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{3 x \sinh ^2(b x)}{2 b^3}+\frac{3 \int \frac{\sinh (2 b x)}{x} \, dx}{b^4}\\ &=-\frac{5 x}{2 b^3}-\frac{x^3}{6 b}+\frac{x \cosh ^2(b x)}{2 b^3}+\frac{6 x \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^3 \cosh (b x) \text{Chi}(b x)}{b}-\frac{4 \cosh (b x) \sinh (b x)}{b^4}-\frac{x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac{6 \text{Chi}(b x) \sinh (b x)}{b^4}-\frac{3 x^2 \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{3 x \sinh ^2(b x)}{2 b^3}+\frac{3 \text{Shi}(2 b x)}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.101521, size = 94, normalized size = 0.64 \[ \frac{12 \text{Chi}(b x) \left (b x \left (b^2 x^2+6\right ) \cosh (b x)-3 \left (b^2 x^2+2\right ) \sinh (b x)\right )-2 b^3 x^3-3 b^2 x^2 \sinh (2 b x)+36 \text{Shi}(2 b x)-36 b x-24 \sinh (2 b x)+12 b x \cosh (2 b x)}{12 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*CoshIntegral[b*x]*Sinh[b*x],x]

[Out]

(-36*b*x - 2*b^3*x^3 + 12*b*x*Cosh[2*b*x] + 12*CoshIntegral[b*x]*(b*x*(6 + b^2*x^2)*Cosh[b*x] - 3*(2 + b^2*x^2
)*Sinh[b*x]) - 24*Sinh[2*b*x] - 3*b^2*x^2*Sinh[2*b*x] + 36*SinhIntegral[2*b*x])/(12*b^4)

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Maple [A]  time = 0.062, size = 104, normalized size = 0.7 \begin{align*}{\frac{1}{{b}^{4}} \left ({\it Chi} \left ( bx \right ) \left ({b}^{3}{x}^{3}\cosh \left ( bx \right ) -3\,{b}^{2}{x}^{2}\sinh \left ( bx \right ) +6\,bx\cosh \left ( bx \right ) -6\,\sinh \left ( bx \right ) \right ) -{\frac{{b}^{2}{x}^{2}\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{2}}-{\frac{{x}^{3}{b}^{3}}{6}}+2\,bx \left ( \cosh \left ( bx \right ) \right ) ^{2}-4\,\cosh \left ( bx \right ) \sinh \left ( bx \right ) -4\,bx+3\,{\it Shi} \left ( 2\,bx \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*Chi(b*x)*sinh(b*x),x)

[Out]

1/b^4*(Chi(b*x)*(b^3*x^3*cosh(b*x)-3*b^2*x^2*sinh(b*x)+6*b*x*cosh(b*x)-6*sinh(b*x))-1/2*b^2*x^2*cosh(b*x)*sinh
(b*x)-1/6*x^3*b^3+2*b*x*cosh(b*x)^2-4*cosh(b*x)*sinh(b*x)-4*b*x+3*Shi(2*b*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*Chi(b*x)*sinh(b*x),x, algorithm="maxima")

[Out]

integrate(x^3*Chi(b*x)*sinh(b*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{Chi}\left (b x\right ) \sinh \left (b x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*Chi(b*x)*sinh(b*x),x, algorithm="fricas")

[Out]

integral(x^3*cosh_integral(b*x)*sinh(b*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sinh{\left (b x \right )} \operatorname{Chi}\left (b x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*Chi(b*x)*sinh(b*x),x)

[Out]

Integral(x**3*sinh(b*x)*Chi(b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*Chi(b*x)*sinh(b*x),x, algorithm="giac")

[Out]

integrate(x^3*Chi(b*x)*sinh(b*x), x)