\(\int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx\) [114]

Optimal result

Integrand size = 12, antiderivative size = 97 \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \operatorname {CosIntegral}(b x)^2-b^2 \operatorname {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \operatorname {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \] Input:

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^3,x]
 

Output:

-1/4*Cos[b*x]^2/x^2 - (Cos[b*x]*CosIntegral[b*x])/(2*x^2) - (b^2*CosIntegr 
al[b*x]^2)/4 - b^2*CosIntegral[2*b*x] + (b*Cos[b*x]*Sin[b*x])/(2*x) + (b*C 
osIntegral[b*x]*Sin[b*x])/(2*x) + (b*Sin[2*b*x])/(4*x)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.31, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {7070, 27, 3042, 3795, 14, 3042, 3793, 2009, 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.

\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x^3} \, dx\)

\(\Big \downarrow \) 7070

\(\displaystyle -\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx+\frac {1}{2} b \int \frac {\cos ^2(b x)}{b x^3}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{2} b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2}dx+\frac {1}{2} \int \frac {\cos ^2(b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3795

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7076

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 7237

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

Input:

Int[(Cos[b*x]*CosIntegral[b*x])/x^3,x]
 

Output:

-1/2*(Cos[b*x]*CosIntegral[b*x])/x^2 + (-1/2*Cos[b*x]^2/x^2 - 2*b^2*(CosIn 
tegral[2*b*x]/2 + Log[x]/2) + b^2*Log[x] + (b*Cos[b*x]*Sin[b*x])/x)/2 - (b 
*((b*CosIntegral[b*x]^2)/2 - (CosIntegral[b*x]*Sin[b*x])/x + (2*b*CosInteg 
ral[2*b*x] - Sin[2*b*x]/x)/2))/2
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[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 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 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 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 3795

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ 
b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) 
*(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2)))   Int[(c + 
d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* 
(m + 2)))   Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, 
c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
 

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 7070

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

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]

\[\int \frac {\cos \left (b x \right ) \operatorname {Ci}\left (b x \right )}{x^{3}}d x\]

Input:

int(cos(b*x)*Ci(b*x)/x^3,x)
 

Output:

int(cos(b*x)*Ci(b*x)/x^3,x)
 

Fricas [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(cos(b*x)*fresnel_cos(b*x)/x^3,x, algorithm="fricas")
 

Output:

integral(cos(b*x)*fresnel_cos(b*x)/x^3, x)
 

Sympy [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int \frac {\cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \] Input:

integrate(cos(b*x)*Ci(b*x)/x**3,x)
 

Output:

Integral(cos(b*x)*Ci(b*x)/x**3, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(cos(b*x)*fresnel_cos(b*x)/x^3,x, algorithm="maxima")
 

Output:

integrate(cos(b*x)*fresnel_cos(b*x)/x^3, x)
 

Giac [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int { \frac {\cos \left (b x\right ) \operatorname {C}\left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(cos(b*x)*fresnel_cos(b*x)/x^3,x, algorithm="giac")
 

Output:

integrate(cos(b*x)*fresnel_cos(b*x)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right )}{x^3} \,d x \] Input:

int((cosint(b*x)*cos(b*x))/x^3,x)
 

Output:

int((cosint(b*x)*cos(b*x))/x^3, x)
 

Reduce [F]

\[ \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^3} \, dx=\int \frac {\mathit {ci} \left (b x \right ) \cos \left (b x \right )}{x^{3}}d x \] Input:

int(cos(b*x)*Ci(b*x)/x^3,x)
 

Output:

int((ci(b*x)*cos(b*x))/x**3,x)