3.59 \(\int x^2 \cosh (a+b x) \text{Shi}(a+b x) \, dx\)

Optimal. Leaf size=219 \[ -\frac{a^2 \text{Chi}(2 a+2 b x)}{2 b^3}+\frac{a^2 \log (a+b x)}{2 b^3}-\frac{\text{Chi}(2 a+2 b x)}{b^3}-\frac{a \text{Shi}(2 a+2 b x)}{b^3}+\frac{2 \text{Shi}(a+b x) \sinh (a+b x)}{b^3}-\frac{2 x \text{Shi}(a+b x) \cosh (a+b x)}{b^2}-\frac{a x}{2 b^2}+\frac{\log (a+b x)}{b^3}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{\cosh (2 a+2 b x)}{2 b^3}+\frac{a \sinh (a+b x) \cosh (a+b x)}{2 b^3}-\frac{x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x^2 \text{Shi}(a+b x) \sinh (a+b x)}{b}+\frac{x^2}{4 b} \]

[Out]

-(a*x)/(2*b^2) + x^2/(4*b) + Cosh[2*a + 2*b*x]/(2*b^3) - CoshIntegral[2*a + 2*b*x]/b^3 - (a^2*CoshIntegral[2*a
+ 2*b*x])/(2*b^3) + Log[a + b*x]/b^3 + (a^2*Log[a + b*x])/(2*b^3) + (a*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) -
(x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + Sinh[a + b*x]^2/(4*b^3) - (2*x*Cosh[a + b*x]*SinhIntegral[a + b*x])
/b^2 + (2*Sinh[a + b*x]*SinhIntegral[a + b*x])/b^3 + (x^2*Sinh[a + b*x]*SinhIntegral[a + b*x])/b - (a*SinhInte
gral[2*a + 2*b*x])/b^3

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Rubi [A]  time = 0.696958, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {6548, 6742, 2635, 8, 3310, 30, 3312, 3301, 6542, 5617, 6741, 2638, 3298, 6546} \[ -\frac{a^2 \text{Chi}(2 a+2 b x)}{2 b^3}+\frac{a^2 \log (a+b x)}{2 b^3}-\frac{\text{Chi}(2 a+2 b x)}{b^3}-\frac{a \text{Shi}(2 a+2 b x)}{b^3}+\frac{2 \text{Shi}(a+b x) \sinh (a+b x)}{b^3}-\frac{2 x \text{Shi}(a+b x) \cosh (a+b x)}{b^2}-\frac{a x}{2 b^2}+\frac{\log (a+b x)}{b^3}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{\cosh (2 a+2 b x)}{2 b^3}+\frac{a \sinh (a+b x) \cosh (a+b x)}{2 b^3}-\frac{x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x^2 \text{Shi}(a+b x) \sinh (a+b x)}{b}+\frac{x^2}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x)/(2*b^2) + x^2/(4*b) + Cosh[2*a + 2*b*x]/(2*b^3) - CoshIntegral[2*a + 2*b*x]/b^3 - (a^2*CoshIntegral[2*a
+ 2*b*x])/(2*b^3) + Log[a + b*x]/b^3 + (a^2*Log[a + b*x])/(2*b^3) + (a*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) -
(x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + Sinh[a + b*x]^2/(4*b^3) - (2*x*Cosh[a + b*x]*SinhIntegral[a + b*x])
/b^2 + (2*Sinh[a + b*x]*SinhIntegral[a + b*x])/b^3 + (x^2*Sinh[a + b*x]*SinhIntegral[a + b*x])/b - (a*SinhInte
gral[2*a + 2*b*x])/b^3

Rule 6548

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 3310

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 30

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

Rule 3312

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 3301

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 6542

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 5617

Int[Cosh[w_]^(p_.)*(u_.)*Sinh[v_]^(p_.), x_Symbol] :> Dist[1/2^p, Int[u*Sinh[2*v]^p, x], x] /; EqQ[w, v] && In
tegerQ[p]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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]

Rule 6546

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]

Rubi steps

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

Mathematica [A]  time = 0.334399, size = 134, normalized size = 0.61 \[ \frac{-4 \left (a^2+2\right ) \text{Chi}(2 (a+b x))+4 a^2 \log (a+b x)+8 \text{Shi}(a+b x) \left (\left (b^2 x^2+2\right ) \sinh (a+b x)-2 b x \cosh (a+b x)\right )-8 a \text{Shi}(2 (a+b x))-4 a b x+8 \log (a+b x)+2 a \sinh (2 (a+b x))-2 b x \sinh (2 (a+b x))+5 \cosh (2 (a+b x))+2 b^2 x^2}{8 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*x + 2*b^2*x^2 + 5*Cosh[2*(a + b*x)] - 4*(2 + a^2)*CoshIntegral[2*(a + b*x)] + 8*Log[a + b*x] + 4*a^2*L
og[a + b*x] + 2*a*Sinh[2*(a + b*x)] - 2*b*x*Sinh[2*(a + b*x)] + 8*(-2*b*x*Cosh[a + b*x] + (2 + b^2*x^2)*Sinh[a
+ b*x])*SinhIntegral[a + b*x] - 8*a*SinhIntegral[2*(a + b*x)])/(8*b^3)

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Maple [A]  time = 0.063, size = 198, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}{\it Shi} \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{b}}-2\,{\frac{x\cosh \left ( bx+a \right ){\it Shi} \left ( bx+a \right ) }{{b}^{2}}}+2\,{\frac{{\it Shi} \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{{b}^{3}}}-{\frac{x\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2\,{b}^{2}}}+{\frac{\cosh \left ( bx+a \right ) a\sinh \left ( bx+a \right ) }{2\,{b}^{3}}}+{\frac{{x}^{2}}{4\,b}}-{\frac{ax}{2\,{b}^{2}}}-{\frac{3\,{a}^{2}}{4\,{b}^{3}}}+{\frac{5\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4\,{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) }{{b}^{3}}}-{\frac{{\it Chi} \left ( 2\,bx+2\,a \right ) }{{b}^{3}}}-{\frac{a{\it Shi} \left ( 2\,bx+2\,a \right ) }{{b}^{3}}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) }{2\,{b}^{3}}}-{\frac{{a}^{2}{\it Chi} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(b*x+a)*Shi(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*Shi(b*x + a)*cosh(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*cosh(a + b*x)*Shi(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*Shi(b*x + a)*cosh(b*x + a), x)