\(\int x^3 \text {Shi}(b x)^2 \, dx\) [10]

Optimal result

Integrand size = 10, antiderivative size = 149 \[ \int x^3 \text {Shi}(b x)^2 \, dx=\frac {x^2}{2 b^2}-\frac {3 \text {Chi}(2 b x)}{2 b^4}+\frac {3 \log (x)}{2 b^4}-\frac {x \cosh (b x) \sinh (b x)}{b^3}+\frac {2 \sinh ^2(b x)}{b^4}+\frac {x^2 \sinh ^2(b x)}{4 b^2}-\frac {3 x \cosh (b x) \text {Shi}(b x)}{b^3}-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 \sinh (b x) \text {Shi}(b x)}{b^4}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6671, 6677, 12, 5480, 3391, 30, 6683, 2644, 6681, 3393, 3382} \[ \int x^3 \text {Shi}(b x)^2 \, dx=-\frac {3 \text {Chi}(2 b x)}{2 b^4}+\frac {3 \text {Shi}(b x) \sinh (b x)}{b^4}+\frac {3 \log (x)}{2 b^4}+\frac {2 \sinh ^2(b x)}{b^4}-\frac {3 x \text {Shi}(b x) \cosh (b x)}{b^3}-\frac {x \sinh (b x) \cosh (b x)}{b^3}+\frac {3 x^2 \text {Shi}(b x) \sinh (b x)}{2 b^2}+\frac {x^2}{2 b^2}+\frac {x^2 \sinh ^2(b x)}{4 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2-\frac {x^3 \text {Shi}(b x) \cosh (b x)}{2 b} \]

[In]

Int[x^3*SinhIntegral[b*x]^2,x]

[Out]

x^2/(2*b^2) - (3*CoshIntegral[2*b*x])/(2*b^4) + (3*Log[x])/(2*b^4) - (x*Cosh[b*x]*Sinh[b*x])/b^3 + (2*Sinh[b*x
]^2)/b^4 + (x^2*Sinh[b*x]^2)/(4*b^2) - (3*x*Cosh[b*x]*SinhIntegral[b*x])/b^3 - (x^3*Cosh[b*x]*SinhIntegral[b*x
])/(2*b) + (3*Sinh[b*x]*SinhIntegral[b*x])/b^4 + (3*x^2*Sinh[b*x]*SinhIntegral[b*x])/(2*b^2) + (x^4*SinhIntegr
al[b*x]^2)/4

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 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

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 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 6671

Int[(x_)^(m_.)*SinhIntegral[(b_.)*(x_)]^2, x_Symbol] :> Simp[x^(m + 1)*(SinhIntegral[b*x]^2/(m + 1)), x] - Dis
t[2/(m + 1), Int[x^m*Sinh[b*x]*SinhIntegral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]

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]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.72 \[ \int x^3 \text {Shi}(b x)^2 \, dx=\frac {3 b^2 x^2+8 \cosh (2 b x)+b^2 x^2 \cosh (2 b x)-12 \text {Chi}(2 b x)+12 \log (x)-4 b x \sinh (2 b x)-4 \left (b x \left (6+b^2 x^2\right ) \cosh (b x)-3 \left (2+b^2 x^2\right ) \sinh (b x)\right ) \text {Shi}(b x)+2 b^4 x^4 \text {Shi}(b x)^2}{8 b^4} \]

[In]

Integrate[x^3*SinhIntegral[b*x]^2,x]

[Out]

(3*b^2*x^2 + 8*Cosh[2*b*x] + b^2*x^2*Cosh[2*b*x] - 12*CoshIntegral[2*b*x] + 12*Log[x] - 4*b*x*Sinh[2*b*x] - 4*
(b*x*(6 + b^2*x^2)*Cosh[b*x] - 3*(2 + b^2*x^2)*Sinh[b*x])*SinhIntegral[b*x] + 2*b^4*x^4*SinhIntegral[b*x]^2)/(
8*b^4)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {\frac {b^{4} x^{4} \operatorname {Shi}\left (b x \right )^{2}}{4}-2 \,\operatorname {Shi}\left (b x \right ) \left (\frac {b^{3} x^{3} \cosh \left (b x \right )}{4}-\frac {3 b^{2} x^{2} \sinh \left (b x \right )}{4}+\frac {3 b x \cosh \left (b x \right )}{2}-\frac {3 \sinh \left (b x \right )}{2}\right )+\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{4}-b x \cosh \left (b x \right ) \sinh \left (b x \right )+\frac {b^{2} x^{2}}{4}+2 \cosh \left (b x \right )^{2}+\frac {3 \ln \left (b x \right )}{2}-\frac {3 \,\operatorname {Chi}\left (2 b x \right )}{2}}{b^{4}}\)	\(120\)
default	\(\frac {\frac {b^{4} x^{4} \operatorname {Shi}\left (b x \right )^{2}}{4}-2 \,\operatorname {Shi}\left (b x \right ) \left (\frac {b^{3} x^{3} \cosh \left (b x \right )}{4}-\frac {3 b^{2} x^{2} \sinh \left (b x \right )}{4}+\frac {3 b x \cosh \left (b x \right )}{2}-\frac {3 \sinh \left (b x \right )}{2}\right )+\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{4}-b x \cosh \left (b x \right ) \sinh \left (b x \right )+\frac {b^{2} x^{2}}{4}+2 \cosh \left (b x \right )^{2}+\frac {3 \ln \left (b x \right )}{2}-\frac {3 \,\operatorname {Chi}\left (2 b x \right )}{2}}{b^{4}}\)	\(120\)

[In]

int(x^3*Shi(b*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int x^3 \text {Shi}(b x)^2 \, dx=\int { x^{3} {\rm Shi}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^3*Shi(b*x)^2,x, algorithm="fricas")

[Out]

integral(x^3*sinh_integral(b*x)^2, x)

Sympy [F]

\[ \int x^3 \text {Shi}(b x)^2 \, dx=\int x^{3} \operatorname {Shi}^{2}{\left (b x \right )}\, dx \]

[In]

integrate(x**3*Shi(b*x)**2,x)

[Out]

Integral(x**3*Shi(b*x)**2, x)

Maxima [F]

\[ \int x^3 \text {Shi}(b x)^2 \, dx=\int { x^{3} {\rm Shi}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^3*Shi(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x^3*Shi(b*x)^2, x)

Giac [F]

\[ \int x^3 \text {Shi}(b x)^2 \, dx=\int { x^{3} {\rm Shi}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^3*Shi(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^3*Shi(b*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \text {Shi}(b x)^2 \, dx=\int x^3\,{\mathrm {sinhint}\left (b\,x\right )}^2 \,d x \]

[In]

int(x^3*sinhint(b*x)^2,x)

[Out]

int(x^3*sinhint(b*x)^2, x)