\(\int x^3 \text {Chi}(b x) \sinh (b x) \, dx\) [120]

Optimal result

Integrand size = 12, antiderivative size = 146 \[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=-\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {3 \text {Shi}(2 b x)}{b^4} \]

[Out]

-5/2*x/b^3-1/6*x^3/b+6*x*Chi(b*x)*cosh(b*x)/b^3+x^3*Chi(b*x)*cosh(b*x)/b+1/2*x*cosh(b*x)^2/b^3+3*Shi(2*b*x)/b^
4-6*Chi(b*x)*sinh(b*x)/b^4-3*x^2*Chi(b*x)*sinh(b*x)/b^2-4*cosh(b*x)*sinh(b*x)/b^4-1/2*x^2*cosh(b*x)*sinh(b*x)/
b^2+3/2*x*sinh(b*x)^2/b^3

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6684, 12, 3392, 30, 2715, 8, 6678, 5480, 6676, 5556, 3379} \[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}+\frac {3 \text {Shi}(2 b x)}{b^4}-\frac {4 \sinh (b x) \cosh (b x)}{b^4}+\frac {6 x \text {Chi}(b x) \cosh (b x)}{b^3}-\frac {5 x}{2 b^3}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {x \cosh ^2(b x)}{2 b^3}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}-\frac {x^2 \sinh (b x) \cosh (b x)}{2 b^2}+\frac {x^3 \text {Chi}(b x) \cosh (b x)}{b}-\frac {x^3}{6 b} \]

[In]

Int[x^3*CoshIntegral[b*x]*Sinh[b*x],x]

[Out]

(-5*x)/(2*b^3) - x^3/(6*b) + (x*Cosh[b*x]^2)/(2*b^3) + (6*x*Cosh[b*x]*CoshIntegral[b*x])/b^3 + (x^3*Cosh[b*x]*
CoshIntegral[b*x])/b - (4*Cosh[b*x]*Sinh[b*x])/b^4 - (x^2*Cosh[b*x]*Sinh[b*x])/(2*b^2) - (6*CoshIntegral[b*x]*
Sinh[b*x])/b^4 - (3*x^2*CoshIntegral[b*x]*Sinh[b*x])/b^2 + (3*x*Sinh[b*x]^2)/(2*b^3) + (3*SinhIntegral[2*b*x])
/b^4

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 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 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 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 5480

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[x^(m - n
+ 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -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 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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {3 \int x^2 \cosh (b x) \text {Chi}(b x) \, dx}{b}-\int \frac {x^2 \cosh ^2(b x)}{b} \, dx \\ & = \frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {6 \int x \text {Chi}(b x) \sinh (b x) \, dx}{b^2}-\frac {\int x^2 \cosh ^2(b x) \, dx}{b}+\frac {3 \int \frac {x \cosh (b x) \sinh (b x)}{b} \, dx}{b} \\ & = \frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}-\frac {\int \cosh ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cosh (b x) \text {Chi}(b x) \, dx}{b^3}+\frac {3 \int x \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac {6 \int \frac {\cosh ^2(b x)}{b} \, dx}{b^2}-\frac {\int x^2 \, dx}{2 b} \\ & = -\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {\cosh (b x) \sinh (b x)}{4 b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}-\frac {\int 1 \, dx}{4 b^3}-\frac {3 \int \sinh ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cosh ^2(b x) \, dx}{b^3}+\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b^3} \\ & = -\frac {x}{4 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b^4}+\frac {3 \int 1 \, dx}{4 b^3}-\frac {3 \int 1 \, dx}{b^3} \\ & = -\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {6 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b^4} \\ & = -\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {3 \int \frac {\sinh (2 b x)}{x} \, dx}{b^4} \\ & = -\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {3 \text {Shi}(2 b x)}{b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.64 \[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=\frac {-36 b x-2 b^3 x^3+12 b x \cosh (2 b x)+12 \text {Chi}(b x) \left (b x \left (6+b^2 x^2\right ) \cosh (b x)-3 \left (2+b^2 x^2\right ) \sinh (b x)\right )-24 \sinh (2 b x)-3 b^2 x^2 \sinh (2 b x)+36 \text {Shi}(2 b x)}{12 b^4} \]

[In]

Integrate[x^3*CoshIntegral[b*x]*Sinh[b*x],x]

[Out]

(-36*b*x - 2*b^3*x^3 + 12*b*x*Cosh[2*b*x] + 12*CoshIntegral[b*x]*(b*x*(6 + b^2*x^2)*Cosh[b*x] - 3*(2 + b^2*x^2
)*Sinh[b*x]) - 24*Sinh[2*b*x] - 3*b^2*x^2*Sinh[2*b*x] + 36*SinhIntegral[2*b*x])/(12*b^4)

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b^{3} x^{3}}{6}+2 b x \cosh \left (b x \right )^{2}-4 \cosh \left (b x \right ) \sinh \left (b x \right )-4 b x +3 \,\operatorname {Shi}\left (2 b x \right )}{b^{4}}\)	\(104\)
default	\(\frac {\operatorname {Chi}\left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b^{3} x^{3}}{6}+2 b x \cosh \left (b x \right )^{2}-4 \cosh \left (b x \right ) \sinh \left (b x \right )-4 b x +3 \,\operatorname {Shi}\left (2 b x \right )}{b^{4}}\)	\(104\)

[In]

int(x^3*Chi(b*x)*sinh(b*x),x,method=_RETURNVERBOSE)

[Out]

1/b^4*(Chi(b*x)*(b^3*x^3*cosh(b*x)-3*b^2*x^2*sinh(b*x)+6*b*x*cosh(b*x)-6*sinh(b*x))-1/2*b^2*x^2*cosh(b*x)*sinh
(b*x)-1/6*b^3*x^3+2*b*x*cosh(b*x)^2-4*cosh(b*x)*sinh(b*x)-4*b*x+3*Shi(2*b*x))

Fricas [F]

\[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=\int { x^{3} {\rm Chi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]

[In]

integrate(x^3*Chi(b*x)*sinh(b*x),x, algorithm="fricas")

[Out]

integral(x^3*cosh_integral(b*x)*sinh(b*x), x)

Sympy [F]

\[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=\int x^{3} \sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \]

[In]

integrate(x**3*Chi(b*x)*sinh(b*x),x)

[Out]

Integral(x**3*sinh(b*x)*Chi(b*x), x)

Maxima [F]

\[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=\int { x^{3} {\rm Chi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]

[In]

integrate(x^3*Chi(b*x)*sinh(b*x),x, algorithm="maxima")

[Out]

integrate(x^3*Chi(b*x)*sinh(b*x), x)

Giac [F]

\[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=\int { x^{3} {\rm Chi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]

[In]

integrate(x^3*Chi(b*x)*sinh(b*x),x, algorithm="giac")

[Out]

integrate(x^3*Chi(b*x)*sinh(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \text {Chi}(b x) \sinh (b x) \, dx=\int x^3\,\mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \]

[In]

int(x^3*coshint(b*x)*sinh(b*x),x)

[Out]

int(x^3*coshint(b*x)*sinh(b*x), x)