\(\int x^3 \sin (b x) \text {Si}(b x) \, dx\) [45]

Optimal result

Integrand size = 12, antiderivative size = 126 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-\frac {x^2}{b^2}-\frac {3 \operatorname {CosIntegral}(2 b x)}{b^4}+\frac {3 \log (x)}{b^4}+\frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {4 \sin ^2(b x)}{b^4}+\frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}-\frac {6 \sin (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-\frac {3 b^2 x^2-8 \cos (2 b x)+b^2 x^2 \cos (2 b x)+12 \operatorname {CosIntegral}(2 b x)-12 \log (x)-4 b x \sin (2 b x)+4 \left (b x \left (-6+b^2 x^2\right ) \cos (b x)-3 \left (-2+b^2 x^2\right ) \sin (b x)\right ) \text {Si}(b x)}{4 b^4} \] Input:

Integrate[x^3*Sin[b*x]*SinIntegral[b*x],x]
 

Output:

-1/4*(3*b^2*x^2 - 8*Cos[2*b*x] + b^2*x^2*Cos[2*b*x] + 12*CosIntegral[2*b*x 
] - 12*Log[x] - 4*b*x*Sin[2*b*x] + 4*(b*x*(-6 + b^2*x^2)*Cos[b*x] - 3*(-2 
+ b^2*x^2)*Sin[b*x])*SinIntegral[b*x])/b^4
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.70, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.750, Rules used = {7067, 27, 3924, 3042, 3791, 15, 7073, 27, 3042, 3791, 15, 7067, 27, 3042, 3044, 15, 7071, 27, 3042, 3793, 2009}

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.

\(\displaystyle \int x^3 \text {Si}(b x) \sin (b x) \, dx\)

\(\Big \downarrow \) 7067

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3924

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

\(\displaystyle \frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\int xdx}{2}+\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}}{b}}{b}+\frac {3 \int x^2 \cos (b x) \text {Si}(b x)dx}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \int x^2 \cos (b x) \text {Si}(b x)dx}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 7073

\(\displaystyle \frac {3 \left (-\frac {2 \int x \sin (b x) \text {Si}(b x)dx}{b}-\int \frac {x \sin ^2(b x)}{b}dx+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (-\frac {2 \int x \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int x \sin ^2(b x)dx}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {2 \int x \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int x \sin (b x)^2dx}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {3 \left (-\frac {\frac {\int xdx}{2}+\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}}{b}-\frac {2 \int x \sin (b x) \text {Si}(b x)dx}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \left (-\frac {2 \int x \sin (b x) \text {Si}(b x)dx}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 7067

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\int \cos (b x) \text {Si}(b x)dx}{b}+\int \frac {\cos (b x) \sin (b x)}{b}dx-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\int \cos (b x) \text {Si}(b x)dx}{b}+\frac {\int \cos (b x) \sin (b x)dx}{b}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\int \cos (b x) \text {Si}(b x)dx}{b}+\frac {\int \cos (b x) \sin (b x)dx}{b}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3044

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\int \sin (b x)d\sin (b x)}{b^2}+\frac {\int \cos (b x) \text {Si}(b x)dx}{b}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\int \cos (b x) \text {Si}(b x)dx}{b}+\frac {\sin ^2(b x)}{2 b^2}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 7071

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\int \frac {\sin ^2(b x)}{b x}dx}{b}+\frac {\sin ^2(b x)}{2 b^2}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin ^2(b x)}{x}dx}{b}}{b}+\frac {\sin ^2(b x)}{2 b^2}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (b x)^2}{x}dx}{b}}{b}+\frac {\sin ^2(b x)}{2 b^2}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right )dx}{b}}{b}+\frac {\sin ^2(b x)}{2 b^2}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (-\frac {2 \left (\frac {\sin ^2(b x)}{2 b^2}+\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\frac {\log (x)}{2}-\frac {\operatorname {CosIntegral}(2 b x)}{2}}{b}}{b}-\frac {x \text {Si}(b x) \cos (b x)}{b}\right )}{b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}\right )}{b}+\frac {\frac {x^2 \sin ^2(b x)}{2 b}-\frac {\frac {\sin ^2(b x)}{4 b^2}-\frac {x \sin (b x) \cos (b x)}{2 b}+\frac {x^2}{4}}{b}}{b}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b}\)

Input:

Int[x^3*Sin[b*x]*SinIntegral[b*x],x]
 

Output:

((x^2*Sin[b*x]^2)/(2*b) - (x^2/4 - (x*Cos[b*x]*Sin[b*x])/(2*b) + Sin[b*x]^ 
2/(4*b^2))/b)/b - (x^3*Cos[b*x]*SinIntegral[b*x])/b + (3*(-((x^2/4 - (x*Co 
s[b*x]*Sin[b*x])/(2*b) + Sin[b*x]^2/(4*b^2))/b) + (x^2*Sin[b*x]*SinIntegra 
l[b*x])/b - (2*(Sin[b*x]^2/(2*b^2) - (x*Cos[b*x]*SinIntegral[b*x])/b + (-( 
(-1/2*CosIntegral[2*b*x] + Log[x]/2)/b) + (Sin[b*x]*SinIntegral[b*x])/b)/b 
))/b))/b
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3044

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ 
Symbol] :> Simp[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] &&  !(I 
ntegerQ[(m - 1)/2] && LtQ[0, m, n])
 

rule 3791

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 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 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 7071

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

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 = 5.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {\operatorname {Si}\left (b x \right ) \left (-b^{3} x^{3} \cos \left (b x \right )+3 b^{2} x^{2} \sin \left (b x \right )-6 \sin \left (b x \right )+6 b x \cos \left (b x \right )\right )-\frac {\cos \left (b x \right )^{2} b^{2} x^{2}}{2}+b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+\frac {b^{2} x^{2}}{2}-\sin \left (b x \right )^{2}-3 b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+3 \cos \left (b x \right )^{2}+3 \ln \left (b x \right )-3 \,\operatorname {Ci}\left (2 b x \right )}{b^{4}}\)	\(138\)
default	\(\frac {\operatorname {Si}\left (b x \right ) \left (-b^{3} x^{3} \cos \left (b x \right )+3 b^{2} x^{2} \sin \left (b x \right )-6 \sin \left (b x \right )+6 b x \cos \left (b x \right )\right )-\frac {\cos \left (b x \right )^{2} b^{2} x^{2}}{2}+b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+\frac {b^{2} x^{2}}{2}-\sin \left (b x \right )^{2}-3 b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+3 \cos \left (b x \right )^{2}+3 \ln \left (b x \right )-3 \,\operatorname {Ci}\left (2 b x \right )}{b^{4}}\)	\(138\)

Input:

int(x^3*sin(b*x)*Si(b*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.73 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-\frac {b^{2} x^{2} + {\left (b^{2} x^{2} - 8\right )} \cos \left (b x\right )^{2} + 2 \, {\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - 2 \, {\left (2 \, b x \cos \left (b x\right ) + 3 \, {\left (b^{2} x^{2} - 2\right )} \operatorname {Si}\left (b x\right )\right )} \sin \left (b x\right ) + 6 \, \operatorname {Ci}\left (2 \, b x\right ) - 6 \, \log \left (x\right )}{2 \, b^{4}} \] Input:

integrate(x^3*sin(b*x)*sin_integral(b*x),x, algorithm="fricas")
 

Output:

-1/2*(b^2*x^2 + (b^2*x^2 - 8)*cos(b*x)^2 + 2*(b^3*x^3 - 6*b*x)*cos(b*x)*si 
n_integral(b*x) - 2*(2*b*x*cos(b*x) + 3*(b^2*x^2 - 2)*sin_integral(b*x))*s 
in(b*x) + 6*cos_integral(2*b*x) - 6*log(x))/b^4
 

Sympy [F]

\[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=\int x^{3} \sin {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \] Input:

integrate(x**3*sin(b*x)*Si(b*x),x)
 

Output:

Integral(x**3*sin(b*x)*Si(b*x), x)
 

Maxima [F]

\[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=\int { x^{3} \sin \left (b x\right ) \operatorname {Si}\left (b x\right ) \,d x } \] Input:

integrate(x^3*sin(b*x)*sin_integral(b*x),x, algorithm="maxima")
 

Output:

integrate(x^3*sin(b*x)*sin_integral(b*x), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.84 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-{\left (\frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right )}{b^{4}} - \frac {3 \, {\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Si}\left (b x\right ) - \frac {b^{2} x^{2} \cos \left (2 \, b x\right ) + 3 \, b^{2} x^{2} - 4 \, b x \sin \left (2 \, b x\right ) - 8 \, \cos \left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (-2 \, b x\right ) - 12 \, \log \left (x\right )}{4 \, b^{4}} \] Input:

integrate(x^3*sin(b*x)*sin_integral(b*x),x, algorithm="giac")
 

Output:

-((b^3*x^3 - 6*b*x)*cos(b*x)/b^4 - 3*(b^2*x^2 - 2)*sin(b*x)/b^4)*sin_integ 
ral(b*x) - 1/4*(b^2*x^2*cos(2*b*x) + 3*b^2*x^2 - 4*b*x*sin(2*b*x) - 8*cos( 
2*b*x) + 6*cos_integral(2*b*x) + 6*cos_integral(-2*b*x) - 12*log(x))/b^4
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=\int x^3\,\mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \] Input:

int(x^3*sinint(b*x)*sin(b*x),x)
 

Output:

int(x^3*sinint(b*x)*sin(b*x), x)
 

Reduce [F]

\[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=\int \mathit {si} \left (b x \right ) \sin \left (b x \right ) x^{3}d x \] Input:

int(x^3*sin(b*x)*Si(b*x),x)
 

Output:

int(si(b*x)*sin(b*x)*x**3,x)