3.128 \(\int x \cosh (a+b x) \text {Chi}(a+b x) \, dx\)

Optimal. Leaf size=97 \[ \frac {\text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b^2}+\frac {a \text {Shi}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}-\frac {\cosh (2 a+2 b x)}{4 b^2}+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6543, 5617, 6741, 6742, 2638, 3298, 6547, 3312, 3301} \[ \frac {\text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b^2}+\frac {a \text {Shi}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}-\frac {\cosh (2 a+2 b x)}{4 b^2}+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[a + b*x]*CoshIntegral[a + b*x],x]

[Out]

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

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 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 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 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 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 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 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]]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 73, normalized size = 0.75 \[ \frac {2 \text {Chi}(2 (a+b x))+4 \text {Chi}(a+b x) (b x \sinh (a+b x)-\cosh (a+b x))+2 a \text {Shi}(2 (a+b x))+2 \log (a+b x)-\cosh (2 (a+b x))}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[a + b*x]*CoshIntegral[a + b*x],x]

[Out]

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

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cosh \left (b x + a\right ) \operatorname {Chi}\left (b x + a\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*cosh(b*x + a)*cosh_integral(b*x + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 89, normalized size = 0.92 \[ \frac {x \Chi \left (b x +a \right ) \sinh \left (b x +a \right )}{b}-\frac {\Chi \left (b x +a \right ) \cosh \left (b x +a \right )}{b^{2}}-\frac {\cosh ^{2}\left (b x +a \right )}{2 b^{2}}+\frac {\ln \left (b x +a \right )}{2 b^{2}}+\frac {\Chi \left (2 b x +2 a \right )}{2 b^{2}}+\frac {a \Shi \left (2 b x +2 a \right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Chi(b*x+a)*cosh(b*x+a),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {coshint}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*coshint(a + b*x)*cosh(a + b*x),x)

[Out]

int(x*coshint(a + b*x)*cosh(a + b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cosh {\left (a + b x \right )} \operatorname {Chi}\left (a + b x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*cosh(a + b*x)*Chi(a + b*x), x)

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