\(\int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx\) [47]

Optimal result

Integrand size = 12, antiderivative size = 44 \[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=b \operatorname {CosIntegral}(2 b x)-\frac {\sin (2 b x)}{2 x}-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2 \] Output:

b*Ci(2*b*x)-1/2*sin(2*b*x)/x-cos(b*x)*Si(b*x)/x-1/2*b*Si(b*x)^2
 

Mathematica [F]

\[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=\int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx \] Input:

Integrate[(Cos[b*x]*SinIntegral[b*x])/x^2,x]
 

Output:

Integrate[(Cos[b*x]*SinIntegral[b*x])/x^2, x]
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7075, 27, 4906, 27, 3042, 3778, 3042, 3783, 7237}

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.

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

\(\Big \downarrow \) 7075

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 7237

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

Input:

Int[(Cos[b*x]*SinIntegral[b*x])/x^2,x]
 

Output:

(2*b*CosIntegral[2*b*x] - Sin[2*b*x]/x)/2 - (Cos[b*x]*SinIntegral[b*x])/x 
- (b*SinIntegral[b*x]^2)/2
 

Defintions of rubi rules used

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 3778

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

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

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 7075

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

rule 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

Maple [F]

\[\int \frac {\cos \left (b x \right ) \operatorname {Si}\left (b x \right )}{x^{2}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx=-\frac {b x \operatorname {Si}\left (b x\right )^{2} - 2 \, b x \operatorname {Ci}\left (2 \, b x\right ) + 2 \, \cos \left (b x\right ) \sin \left (b x\right ) + 2 \, \cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{2 \, x} \] Input:

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

Output:

-1/2*(b*x*sin_integral(b*x)^2 - 2*b*x*cos_integral(2*b*x) + 2*cos(b*x)*sin 
(b*x) + 2*cos(b*x)*sin_integral(b*x))/x
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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