\(\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx\) [108]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\frac {1}{2} b \operatorname {CosIntegral}(b x)^2+b \operatorname {CosIntegral}(2 b x)-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x} \] Input:

Integrate[(CosIntegral[b*x]*Sin[b*x])/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.54 (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 = {7076, 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.

\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx\)

\(\Big \downarrow \) 7076

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 7237

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

Input:

Int[(CosIntegral[b*x]*Sin[b*x])/x^2,x]
 

Output:

(b*CosIntegral[b*x]^2)/2 - (CosIntegral[b*x]*Sin[b*x])/x + (2*b*CosIntegra 
l[2*b*x] - Sin[2*b*x]/x)/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 7076

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

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

Input:

int(Ci(b*x)*sin(b*x)/x^2,x)
 

Output:

int(Ci(b*x)*sin(b*x)/x^2,x)
 

Fricas [F]

\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(fresnel_cos(b*x)*sin(b*x)/x^2,x, algorithm="fricas")
 

Output:

integral(fresnel_cos(b*x)*sin(b*x)/x^2, x)
 

Sympy [F]

\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int \frac {\sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{2}}\, dx \] Input:

integrate(Ci(b*x)*sin(b*x)/x**2,x)
 

Output:

Integral(sin(b*x)*Ci(b*x)/x**2, x)
 

Maxima [F]

\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(fresnel_cos(b*x)*sin(b*x)/x^2,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos(b*x)*sin(b*x)/x^2, x)
 

Giac [F]

\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{2}} \,d x } \] Input:

integrate(fresnel_cos(b*x)*sin(b*x)/x^2,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x)*sin(b*x)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^2} \,d x \] Input:

int((cosint(b*x)*sin(b*x))/x^2,x)
 

Output:

int((cosint(b*x)*sin(b*x))/x^2, x)
 

Reduce [F]

\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int \frac {\mathit {ci} \left (b x \right ) \sin \left (b x \right )}{x^{2}}d x \] Input:

int(Ci(b*x)*sin(b*x)/x^2,x)
 

Output:

int((ci(b*x)*sin(b*x))/x**2,x)