\(\int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx\) [59]

Optimal result

Integrand size = 16, antiderivative size = 219 \[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=-\frac {a x}{2 b^2}+\frac {x^2}{4 b}+\frac {\cosh (2 a+2 b x)}{2 b^3}-\frac {\text {Chi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Chi}(2 a+2 b x)}{2 b^3}+\frac {\log (a+b x)}{b^3}+\frac {a^2 \log (a+b x)}{2 b^3}+\frac {a \cosh (a+b x) \sinh (a+b x)}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}-\frac {2 x \cosh (a+b x) \text {Shi}(a+b x)}{b^2}+\frac {2 \sinh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \sinh (a+b x) \text {Shi}(a+b x)}{b}-\frac {a \text {Shi}(2 a+2 b x)}{b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6683, 6874, 2715, 8, 3391, 30, 3393, 3382, 6677, 5736, 6873, 2718, 3379, 6681} \[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=-\frac {a^2 \text {Chi}(2 a+2 b x)}{2 b^3}+\frac {a^2 \log (a+b x)}{2 b^3}-\frac {\text {Chi}(2 a+2 b x)}{b^3}-\frac {a \text {Shi}(2 a+2 b x)}{b^3}+\frac {2 \text {Shi}(a+b x) \sinh (a+b x)}{b^3}+\frac {\log (a+b x)}{b^3}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {\cosh (2 a+2 b x)}{2 b^3}+\frac {a \sinh (a+b x) \cosh (a+b x)}{2 b^3}-\frac {2 x \text {Shi}(a+b x) \cosh (a+b x)}{b^2}-\frac {a x}{2 b^2}-\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x^2 \text {Shi}(a+b x) \sinh (a+b x)}{b}+\frac {x^2}{4 b} \]

[In]

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

[Out]

-1/2*(a*x)/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 8

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

Rule 30

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

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 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 3391

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 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 6677

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 6681

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]

Rule 6683

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 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^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] (verified)

Time = 0.22 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.61 \[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=\frac {-4 a b x+2 b^2 x^2+5 \cosh (2 (a+b x))-4 \left (2+a^2\right ) \text {Chi}(2 (a+b x))+8 \log (a+b x)+4 a^2 \log (a+b x)+2 a \sinh (2 (a+b x))-2 b x \sinh (2 (a+b x))+8 \left (-2 b x \cosh (a+b x)+\left (2+b^2 x^2\right ) \sinh (a+b x)\right ) \text {Shi}(a+b x)-8 a \text {Shi}(2 (a+b x))}{8 b^3} \]

[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)

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (a^{2} \sinh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )+\frac {a^{2} \ln \left (b x +a \right )}{2}-\frac {a^{2} \operatorname {Chi}\left (2 b x +2 a \right )}{2}+\cosh \left (b x +a \right ) \sinh \left (b x +a \right ) a -\left (b x +a \right ) a -a \,\operatorname {Shi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}+\frac {5 \cosh \left (b x +a \right )^{2}}{4}+\ln \left (b x +a \right )-\operatorname {Chi}\left (2 b x +2 a \right )}{b^{3}}\)	\(197\)
default	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (a^{2} \sinh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )+\frac {a^{2} \ln \left (b x +a \right )}{2}-\frac {a^{2} \operatorname {Chi}\left (2 b x +2 a \right )}{2}+\cosh \left (b x +a \right ) \sinh \left (b x +a \right ) a -\left (b x +a \right ) a -a \,\operatorname {Shi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}+\frac {5 \cosh \left (b x +a \right )^{2}}{4}+\ln \left (b x +a \right )-\operatorname {Chi}\left (2 b x +2 a \right )}{b^{3}}\)	\(197\)

[In]

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

[Out]

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

Fricas [F]

\[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]

[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)

Sympy [F]

\[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=\int x^{2} \cosh {\left (a + b x \right )} \operatorname {Shi}{\left (a + b x \right )}\, dx \]

[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)

Maxima [F]

\[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]

[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)

Giac [F]

\[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x^2 \cosh (a+b x) \text {Shi}(a+b x) \, dx=\int x^2\,\mathrm {sinhint}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]

[In]

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

[Out]

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