\(\int x \cosh (a+b x) \text {Chi}(a+b x) \, dx\) [128]

Optimal result

Integrand size = 14, antiderivative size = 97 \[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=-\frac {\cosh (2 a+2 b x)}{4 b^2}-\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b^2}+\frac {\text {Chi}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {a \text {Shi}(2 a+2 b x)}{2 b^2} \]

[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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6678, 5736, 6873, 6874, 2718, 3379, 6682, 3393, 3382} \[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\frac {\text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b^2}+\frac {a \text {Shi}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}-\frac {\cosh (2 a+2 b x)}{4 b^2}+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b} \]

[In]

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

[Out]

-1/4*Cosh[2*a + 2*b*x]/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 2718

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

Rule 3379

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 3382

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 3393

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 5736

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 6678

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 6682

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 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\frac {-\cosh (2 (a+b x))+2 \text {Chi}(2 (a+b x))+2 \log (a+b x)+4 \text {Chi}(a+b x) (-\cosh (a+b x)+b x \sinh (a+b x))+2 a \text {Shi}(2 (a+b x))}{4 b^2} \]

[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)

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\frac {a \,\operatorname {Shi}\left (2 b x +2 a \right )}{2}-\frac {\cosh \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(84\)
default	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\frac {a \,\operatorname {Shi}\left (2 b x +2 a \right )}{2}-\frac {\cosh \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(84\)

[In]

int(x*Chi(b*x+a)*cosh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]

[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)

Sympy [F]

\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int x \cosh {\left (a + b x \right )} \operatorname {Chi}\left (a + b x\right )\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]

[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)

Giac [F]

\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int x\,\mathrm {coshint}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]

[In]

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

[Out]

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