\(\int x^2 \sinh (b x) \text {Shi}(b x) \, dx\) [44]

Optimal result

Integrand size = 12, antiderivative size = 90 \[ \int x^2 \sinh (b x) \text {Shi}(b x) \, dx=-\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {\text {Shi}(2 b x)}{b^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.71 \[ \int x^2 \sinh (b x) \text {Shi}(b x) \, dx=\frac {-8 b x-2 b x \cosh (2 b x)+5 \sinh (2 b x)+8 \left (\left (2+b^2 x^2\right ) \cosh (b x)-2 b x \sinh (b x)\right ) \text {Shi}(b x)-8 \text {Shi}(2 b x)}{8 b^3} \] Input:

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

Output:

(-8*b*x - 2*b*x*Cosh[2*b*x] + 5*Sinh[2*b*x] + 8*((2 + b^2*x^2)*Cosh[b*x] - 
 2*b*x*Sinh[b*x])*SinhIntegral[b*x] - 8*SinhIntegral[2*b*x])/(8*b^3)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.53, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.667, Rules used = {7096, 27, 5895, 3042, 25, 3115, 24, 7102, 27, 3042, 25, 3115, 24, 7094, 27, 5971, 27, 3042, 26, 3779}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \text {Shi}(b x) \sinh (b x) \, dx\)

\(\Big \downarrow \) 7096

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Shi}(b x)dx}{b}-\int \frac {x \cosh (b x) \sinh (b x)}{b}dx+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Shi}(b x)dx}{b}-\frac {\int x \cosh (b x) \sinh (b x)dx}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 5895

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Shi}(b x)dx}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}-\frac {\int \sinh ^2(b x)dx}{2 b}}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Shi}(b x)dx}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}-\frac {\int -\sin (i b x)^2dx}{2 b}}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Shi}(b x)dx}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\int \sin (i b x)^2dx}{2 b}}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Shi}(b x)dx}{b}-\frac {\frac {\frac {\int 1dx}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}+\frac {x \sinh ^2(b x)}{2 b}}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Shi}(b x)dx}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 7102

\(\displaystyle -\frac {2 \left (-\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\int \frac {\sinh ^2(b x)}{b}dx+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\frac {\int \sinh ^2(b x)dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (-\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}-\frac {\int -\sin (i b x)^2dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (-\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {\int \sin (i b x)^2dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {2 \left (-\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {\frac {\int 1dx}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {2 \left (-\frac {\int \sinh (b x) \text {Shi}(b x)dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 7094

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b x}dx}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\cosh (b x) \sinh (b x)}{x}dx}{b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{2 x}dx}{b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sinh (2 b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\int -\frac {i \sin (2 i b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}+\frac {i \int \frac {\sin (2 i b x)}{x}dx}{2 b}}{b}+\frac {x \text {Shi}(b x) \sinh (b x)}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {x^2 \text {Shi}(b x) \cosh (b x)}{b}-\frac {2 \left (\frac {x \text {Shi}(b x) \sinh (b x)}{b}-\frac {\frac {\text {Shi}(b x) \cosh (b x)}{b}-\frac {\text {Shi}(2 b x)}{2 b}}{b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{b}\right )}{b}-\frac {\frac {x \sinh ^2(b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (b x) \cosh (b x)}{2 b}}{2 b}}{b}\)

Input:

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

Output:

-(((x*Sinh[b*x]^2)/(2*b) + (x/2 - (Cosh[b*x]*Sinh[b*x])/(2*b))/(2*b))/b) + 
 (x^2*Cosh[b*x]*SinhIntegral[b*x])/b - (2*((x/2 - (Cosh[b*x]*Sinh[b*x])/(2 
*b))/b + (x*Sinh[b*x]*SinhIntegral[b*x])/b - ((Cosh[b*x]*SinhIntegral[b*x] 
)/b - SinhIntegral[2*b*x]/(2*b))/b))/b
 

Defintions of rubi rules used

rule 24

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 5895

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] - Simp[(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 5971

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 7094

Int[Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> 
 Simp[Cosh[a + b*x]*(SinhIntegral[c + d*x]/b), x] - Simp[d/b   Int[Cosh[a + 
 b*x]*(Sinh[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
 

rule 7096

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] + (-Simp[d/b   Int[(e + f*x)^m*Cosh[a + b*x]*(Sinh[c + d*x]/( 
c + d*x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*Sinh 
Integral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

rule 7102

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] + (-Simp[d/b   Int[(e + f*x)^m*Sinh[a + b*x]*(Sinh[c + d*x]/( 
c + d*x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Sinh[a + b*x]*Sinh 
Integral[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right )^{2}}{2}+\frac {5 \cosh \left (b x \right ) \sinh \left (b x \right )}{4}-\frac {3 b x}{4}-\operatorname {Shi}\left (2 b x \right )}{b^{3}}\)	\(68\)
default	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right )^{2}}{2}+\frac {5 \cosh \left (b x \right ) \sinh \left (b x \right )}{4}-\frac {3 b x}{4}-\operatorname {Shi}\left (2 b x \right )}{b^{3}}\)	\(68\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x^2*sinh(b*x)*sinh_integral(b*x), x)
 

Sympy [F]

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

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

Output:

Integral(x**2*sinh(b*x)*Shi(b*x), x)
 

Maxima [F]

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

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

Output:

integrate(x^2*Shi(b*x)*sinh(b*x), x)
 

Giac [F]

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

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

Output:

integrate(x^2*Shi(b*x)*sinh(b*x), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \sinh (b x) \text {Shi}(b x) \, dx=\int \mathit {shi} \left (b x \right ) \sinh \left (b x \right ) x^{2}d x \] Input:

int(x^2*sinh(b*x)*Shi(b*x),x)
 

Output:

int(shi(b*x)*sinh(b*x)*x**2,x)