\(\int x^2 \text {Si}(b x)^2 \, dx\) [11]

Optimal result

Integrand size = 10, antiderivative size = 112 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {5 x}{6 b^2}-\frac {5 \cos (b x) \sin (b x)}{6 b^3}-\frac {x \sin ^2(b x)}{3 b^2}-\frac {4 \cos (b x) \text {Si}(b x)}{3 b^3}+\frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {2 \text {Si}(2 b x)}{3 b^3} \] Output:

5/6*x/b^2-5/6*cos(b*x)*sin(b*x)/b^3-1/3*x*sin(b*x)^2/b^2-4/3*cos(b*x)*Si(b 
*x)/b^3+2/3*x^2*cos(b*x)*Si(b*x)/b-4/3*x*sin(b*x)*Si(b*x)/b^2+1/3*x^3*Si(b 
*x)^2+2/3*Si(2*b*x)/b^3
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.70 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {8 b x+2 b x \cos (2 b x)-5 \sin (2 b x)+8 \left (\left (-2+b^2 x^2\right ) \cos (b x)-2 b x \sin (b x)\right ) \text {Si}(b x)+4 b^3 x^3 \text {Si}(b x)^2+8 \text {Si}(2 b x)}{12 b^3} \] Input:

Integrate[x^2*SinIntegral[b*x]^2,x]
 

Output:

(8*b*x + 2*b*x*Cos[2*b*x] - 5*Sin[2*b*x] + 8*((-2 + b^2*x^2)*Cos[b*x] - 2* 
b*x*Sin[b*x])*SinIntegral[b*x] + 4*b^3*x^3*SinIntegral[b*x]^2 + 8*SinInteg 
ral[2*b*x])/(12*b^3)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.41, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.800, Rules used = {7061, 7067, 27, 3924, 3042, 3115, 24, 7073, 27, 3042, 3115, 24, 7065, 27, 4906, 27, 3042, 3780}

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.

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

\(\Big \downarrow \) 7061

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \int x^2 \sin (b x) \text {Si}(b x)dx\)

\(\Big \downarrow \) 7067

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\int \frac {x \cos (b x) \sin (b x)}{b}dx-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\int x \cos (b x) \sin (b x)dx}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )\)

\(\Big \downarrow \) 3924

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\int \sin ^2(b x)dx}{2 b}}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\int \sin (b x)^2dx}{2 b}}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {\int 1dx}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \int x \cos (b x) \text {Si}(b x)dx}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 7073

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\int \frac {\sin ^2(b x)}{b}dx+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int \sin ^2(b x)dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int \sin (b x)^2dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\frac {\int 1dx}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\int \sin (b x) \text {Si}(b x)dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 7065

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\int \frac {\cos (b x) \sin (b x)}{b x}dx-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\frac {\int \frac {\cos (b x) \sin (b x)}{x}dx}{b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\frac {\int \frac {\sin (2 b x)}{2 x}dx}{b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (\frac {2 \left (-\frac {\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \left (-\frac {x^2 \text {Si}(b x) \cos (b x)}{b}+\frac {2 \left (\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {\text {Si}(2 b x)}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\right )}{b}+\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\right )\)

Input:

Int[x^2*SinIntegral[b*x]^2,x]
 

Output:

(x^3*SinIntegral[b*x]^2)/3 - (2*(((x*Sin[b*x]^2)/(2*b) - (x/2 - (Cos[b*x]* 
Sin[b*x])/(2*b))/(2*b))/b - (x^2*Cos[b*x]*SinIntegral[b*x])/b + (2*(-((x/2 
 - (Cos[b*x]*Sin[b*x])/(2*b))/b) + (x*Sin[b*x]*SinIntegral[b*x])/b - (-((C 
os[b*x]*SinIntegral[b*x])/b) + SinIntegral[2*b*x]/(2*b))/b))/b))/3
 

Defintions of rubi rules used

rule 24

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

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 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3924

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

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 7061

Int[(x_)^(m_.)*SinIntegral[(b_.)*(x_)]^2, x_Symbol] :> Simp[x^(m + 1)*(SinI 
ntegral[b*x]^2/(m + 1)), x] - Simp[2/(m + 1)   Int[x^m*Sin[b*x]*SinIntegral 
[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
 

rule 7065

Int[Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[(-Cos[a + b*x])*(SinIntegral[c + d*x]/b), x] + Simp[d/b   Int[Cos[a + b 
*x]*(Sin[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
 

rule 7067

Int[((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + 
(d_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*Cos[a + b*x]*(SinIntegral[c + 
 d*x]/b), x] + (Simp[d/b   Int[(e + f*x)^m*Cos[a + b*x]*(Sin[c + d*x]/(c + 
d*x)), x], x] + Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Cos[a + b*x]*SinIntegr 
al[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

rule 7073

Int[Cos[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinIntegral[(c_.) + 
(d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^m*Sin[a + b*x]*(SinIntegral[c + d* 
x]/b), x] + (-Simp[d/b   Int[(e + f*x)^m*Sin[a + b*x]*(Sin[c + d*x]/(c + d* 
x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Sin[a + b*x]*SinIntegral 
[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 7.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {\frac {b^{3} x^{3} \operatorname {Si}\left (b x \right )^{2}}{3}-2 \,\operatorname {Si}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (b x \right )}{3}+\frac {2 \cos \left (b x \right )}{3}+\frac {2 b x \sin \left (b x \right )}{3}\right )+\frac {\cos \left (b x \right )^{2} b x}{3}-\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{6}+\frac {b x}{2}+\frac {2 \,\operatorname {Si}\left (2 b x \right )}{3}}{b^{3}}\)	\(84\)
default	\(\frac {\frac {b^{3} x^{3} \operatorname {Si}\left (b x \right )^{2}}{3}-2 \,\operatorname {Si}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (b x \right )}{3}+\frac {2 \cos \left (b x \right )}{3}+\frac {2 b x \sin \left (b x \right )}{3}\right )+\frac {\cos \left (b x \right )^{2} b x}{3}-\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{6}+\frac {b x}{2}+\frac {2 \,\operatorname {Si}\left (2 b x \right )}{3}}{b^{3}}\)	\(84\)

Input:

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

Output:

1/b^3*(1/3*b^3*x^3*Si(b*x)^2-2*Si(b*x)*(-1/3*b^2*x^2*cos(b*x)+2/3*cos(b*x) 
+2/3*b*x*sin(b*x))+1/3*cos(b*x)^2*b*x-5/6*sin(b*x)*cos(b*x)+1/2*b*x+2/3*Si 
(2*b*x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {2 \, b^{3} x^{3} \operatorname {Si}\left (b x\right )^{2} + 2 \, b x \cos \left (b x\right )^{2} + 4 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) + 3 \, b x - {\left (8 \, b x \operatorname {Si}\left (b x\right ) + 5 \, \cos \left (b x\right )\right )} \sin \left (b x\right ) + 4 \, \operatorname {Si}\left (2 \, b x\right )}{6 \, b^{3}} \] Input:

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

Output:

1/6*(2*b^3*x^3*sin_integral(b*x)^2 + 2*b*x*cos(b*x)^2 + 4*(b^2*x^2 - 2)*co 
s(b*x)*sin_integral(b*x) + 3*b*x - (8*b*x*sin_integral(b*x) + 5*cos(b*x))* 
sin(b*x) + 4*sin_integral(2*b*x))/b^3
 

Sympy [F]

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

integrate(x**2*Si(b*x)**2,x)
 

Output:

Integral(x**2*Si(b*x)**2, x)
 

Maxima [F]

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

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

Output:

integrate(x^2*sin_integral(b*x)^2, x)
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.34 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {1}{3} \, x^{3} \operatorname {Si}\left (b x\right )^{2} - \frac {2}{3} \, {\left (\frac {2 \, x \sin \left (b x\right )}{b^{2}} - \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{3}}\right )} \operatorname {Si}\left (b x\right ) + \frac {3 \, b x \tan \left (b x\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + 4 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} + 5 \, b x + 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) + 4 \, \operatorname {Si}\left (2 \, b x\right ) - 5 \, \tan \left (b x\right )}{6 \, {\left (b^{3} \tan \left (b x\right )^{2} + b^{3}\right )}} \] Input:

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

Output:

1/3*x^3*sin_integral(b*x)^2 - 2/3*(2*x*sin(b*x)/b^2 - (b^2*x^2 - 2)*cos(b* 
x)/b^3)*sin_integral(b*x) + 1/6*(3*b*x*tan(b*x)^2 + 2*imag_part(cos_integr 
al(2*b*x))*tan(b*x)^2 - 2*imag_part(cos_integral(-2*b*x))*tan(b*x)^2 + 4*s 
in_integral(2*b*x)*tan(b*x)^2 + 5*b*x + 2*imag_part(cos_integral(2*b*x)) - 
 2*imag_part(cos_integral(-2*b*x)) + 4*sin_integral(2*b*x) - 5*tan(b*x))/( 
b^3*tan(b*x)^2 + b^3)
 

Mupad [F(-1)]

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

int(x^2*sinint(b*x)^2,x)
 

Output:

int(x^2*sinint(b*x)^2, x)
 

Reduce [F]

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

int(x^2*Si(b*x)^2,x)
 

Output:

int(si(b*x)**2*x**2,x)