3.1.51 \(\int x^2 \cos (b x) \text {Si}(b x) \, dx\) [51]

3.1.51.1 Optimal result

Integrand size = 12, antiderivative size = 98 \[ \int x^2 \cos (b x) \text {Si}(b x) \, dx=-\frac {x^2}{4 b}-\frac {\operatorname {CosIntegral}(2 b x)}{b^3}+\frac {\log (x)}{b^3}+\frac {x \cos (b x) \sin (b x)}{2 b^2}-\frac {5 \sin ^2(b x)}{4 b^3}+\frac {2 x \cos (b x) \text {Si}(b x)}{b^2}-\frac {2 \sin (b x) \text {Si}(b x)}{b^3}+\frac {x^2 \sin (b x) \text {Si}(b x)}{b} \]

output

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

3.1.51.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.73 \[ \int x^2 \cos (b x) \text {Si}(b x) \, dx=\frac {-2 b^2 x^2+5 \cos (2 b x)-8 \operatorname {CosIntegral}(2 b x)+8 \log (x)+2 b x \sin (2 b x)+8 \left (2 b x \cos (b x)+\left (-2+b^2 x^2\right ) \sin (b x)\right ) \text {Si}(b x)}{8 b^3} \]

input

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

output

(-2*b^2*x^2 + 5*Cos[2*b*x] - 8*CosIntegral[2*b*x] + 8*Log[x] + 2*b*x*Sin[2 
*b*x] + 8*(2*b*x*Cos[b*x] + (-2 + b^2*x^2)*Sin[b*x])*SinIntegral[b*x])/(8* 
b^3)
 

3.1.51.3 Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {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.

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

\(\Big \downarrow \) 7073

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7067

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3044

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7071

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

input

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

output

-((x^2/4 - (x*Cos[b*x]*Sin[b*x])/(2*b) + Sin[b*x]^2/(4*b^2))/b) + (x^2*Sin 
[b*x]*SinIntegral[b*x])/b - (2*(Sin[b*x]^2/(2*b^2) - (x*Cos[b*x]*SinIntegr 
al[b*x])/b + (-((-1/2*CosIntegral[2*b*x] + Log[x]/2)/b) + (Sin[b*x]*SinInt 
egral[b*x])/b)/b))/b
 

3.1.51.3.1 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 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]
 

3.1.51.4 Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\operatorname {Si}\left (b x \right ) \left (b^{2} x^{2} \sin \left (b x \right )-2 \sin \left (b x \right )+2 b x \cos \left (b x \right )\right )-b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+\frac {b^{2} x^{2}}{4}-\frac {\sin \left (b x \right )^{2}}{4}+\ln \left (b x \right )-\operatorname {Ci}\left (2 b x \right )+\cos \left (b x \right )^{2}}{b^{3}}\)	\(89\)
default	\(\frac {\operatorname {Si}\left (b x \right ) \left (b^{2} x^{2} \sin \left (b x \right )-2 \sin \left (b x \right )+2 b x \cos \left (b x \right )\right )-b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+\frac {b^{2} x^{2}}{4}-\frac {\sin \left (b x \right )^{2}}{4}+\ln \left (b x \right )-\operatorname {Ci}\left (2 b x \right )+\cos \left (b x \right )^{2}}{b^{3}}\)	\(89\)

input

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

output

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

3.1.51.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74 \[ \int x^2 \cos (b x) \text {Si}(b x) \, dx=-\frac {b^{2} x^{2} - 8 \, b x \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - 5 \, \cos \left (b x\right )^{2} - 2 \, {\left (b x \cos \left (b x\right ) + 2 \, {\left (b^{2} x^{2} - 2\right )} \operatorname {Si}\left (b x\right )\right )} \sin \left (b x\right ) + 4 \, \operatorname {Ci}\left (2 \, b x\right ) - 4 \, \log \left (x\right )}{4 \, b^{3}} \]

input

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

output

-1/4*(b^2*x^2 - 8*b*x*cos(b*x)*sin_integral(b*x) - 5*cos(b*x)^2 - 2*(b*x*c 
os(b*x) + 2*(b^2*x^2 - 2)*sin_integral(b*x))*sin(b*x) + 4*cos_integral(2*b 
*x) - 4*log(x))/b^3
 

3.1.51.6 Sympy [F]

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

input

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

output

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

3.1.51.7 Maxima [F]

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

input

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

output

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

3.1.51.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int x^2 \cos (b x) \text {Si}(b x) \, dx={\left (\frac {2 \, x \cos \left (b x\right )}{b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{3}}\right )} \operatorname {Si}\left (b x\right ) - \frac {2 \, b^{2} x^{2} - 2 \, b x \sin \left (2 \, b x\right ) - 5 \, \cos \left (2 \, b x\right ) + 4 \, \operatorname {Ci}\left (2 \, b x\right ) + 4 \, \operatorname {Ci}\left (-2 \, b x\right ) - 8 \, \log \left (x\right )}{8 \, b^{3}} \]

input

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

output

(2*x*cos(b*x)/b^2 + (b^2*x^2 - 2)*sin(b*x)/b^3)*sin_integral(b*x) - 1/8*(2 
*b^2*x^2 - 2*b*x*sin(2*b*x) - 5*cos(2*b*x) + 4*cos_integral(2*b*x) + 4*cos 
_integral(-2*b*x) - 8*log(x))/b^3
 

3.1.51.9 Mupad [F(-1)]

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

input

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

output

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