\(\int x \text {Chi}(a+b x) \sinh (a+b x) \, dx\) [124]

Optimal result

Integrand size = 14, antiderivative size = 109 \[ \int x \text {Chi}(a+b x) \sinh (a+b x) \, dx=-\frac {x}{2 b}+\frac {x \cosh (a+b x) \text {Chi}(a+b x)}{b}+\frac {a \text {Chi}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}-\frac {\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6684, 6874, 2715, 8, 3393, 3382, 6676, 5556, 12, 3379} \[ \int x \text {Chi}(a+b x) \sinh (a+b x) \, dx=\frac {a \text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x \text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {x}{2 b} \]

[In]

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

[Out]

-1/2*x/b + (x*Cosh[a + b*x]*CoshIntegral[a + b*x])/b + (a*CoshIntegral[2*a + 2*b*x])/(2*b^2) + (a*Log[a + b*x]
)/(2*b^2) - (Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) - (CoshIntegral[a + b*x]*Sinh[a + b*x])/b^2 + SinhIntegral[2
*a + 2*b*x]/(2*b^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 2715

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] && Integ
erQ[2*n]

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 5556

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 6676

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 6684

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 6874

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

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int x \text {Chi}(a+b x) \sinh (a+b x) \, dx=\frac {-2 b x+2 a \text {Chi}(2 (a+b x))+2 a \log (a+b x)+4 \text {Chi}(a+b x) (b x \cosh (a+b x)-\sinh (a+b x))-\sinh (2 (a+b x))+2 \text {Shi}(2 (a+b x))}{4 b^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (-a \cosh \left (b x +a \right )+\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+a \left (\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}\right )-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}+\frac {\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(97\)
default	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (-a \cosh \left (b x +a \right )+\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+a \left (\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}\right )-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}+\frac {\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(97\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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