3.95 \(\int x \text{Chi}(a+b x)^2 \, dx\)

Optimal. Leaf size=154 \[ -\frac{a (a+b x) \text{Chi}(a+b x)^2}{2 b^2}-\frac{\text{Chi}(2 a+2 b x)}{2 b^2}+\frac{a \text{Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac{\text{Chi}(a+b x) \cosh (a+b x)}{b^2}-\frac{a \text{Shi}(2 a+2 b x)}{b^2}-\frac{\log (a+b x)}{2 b^2}+\frac{\cosh (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{Chi}(a+b x)^2}{2 b}-\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.32728, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.4, Rules used = {6539, 6543, 5617, 6741, 6742, 2638, 3298, 6547, 3312, 3301, 6535, 6541, 5448, 12} \[ -\frac{a (a+b x) \text{Chi}(a+b x)^2}{2 b^2}-\frac{\text{Chi}(2 a+2 b x)}{2 b^2}+\frac{a \text{Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac{\text{Chi}(a+b x) \cosh (a+b x)}{b^2}-\frac{a \text{Shi}(2 a+2 b x)}{b^2}-\frac{\log (a+b x)}{2 b^2}+\frac{\cosh (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{Chi}(a+b x)^2}{2 b}-\frac{x \text{Chi}(a+b x) \sinh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[x*CoshIntegral[a + b*x]^2,x]

[Out]

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

Rule 6539

Int[CoshIntegral[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)*(c + d*x)^m*Cos
hIntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Cosh[a + b*x]*CoshIntegral[a + b*x],
x], x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*CoshIntegral[a + b*x]^2, x], x]) /; FreeQ[{a
, b, c, d}, 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 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 6742

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

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 6547

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

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 6535

Int[CoshIntegral[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*CoshIntegral[a + b*x]^2)/b, x] - Dist[2,
Int[Cosh[a + b*x]*CoshIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]

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 12

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

Rubi steps

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

Mathematica [A]  time = 0.230468, size = 95, normalized size = 0.62 \[ \frac{-2 \left (a^2-b^2 x^2\right ) \text{Chi}(a+b x)^2-2 \text{Chi}(2 (a+b x))+4 \text{Chi}(a+b x) ((a-b x) \sinh (a+b x)+\cosh (a+b x))-4 a \text{Shi}(2 (a+b x))-2 \log (a+b x)+\cosh (2 (a+b x))}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.059, size = 135, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ({\it Chi} \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{ \left ({\it Chi} \left ( bx+a \right ) \right ) ^{2}{a}^{2}}{2\,{b}^{2}}}-{\frac{x{\it Chi} \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{b}}+{\frac{{\it Chi} \left ( bx+a \right ) a\sinh \left ( bx+a \right ) }{{b}^{2}}}+{\frac{{\it Chi} \left ( bx+a \right ) \cosh \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( bx+a \right ) }{2\,{b}^{2}}}-{\frac{{\it Chi} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{2}}}-{\frac{a{\it Shi} \left ( 2\,bx+2\,a \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Chi(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*cosh_integral(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*Chi(a + b*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Chi(b*x + a)^2, x)