\(\int x^3 \cos (b x) \operatorname {CosIntegral}(b x) \, dx\) [120]

Optimal result

Integrand size = 12, antiderivative size = 142 \[ \int x^3 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=-\frac {x^2}{2 b^2}-\frac {3 \cos ^2(b x)}{4 b^4}-\frac {6 \cos (b x) \operatorname {CosIntegral}(b x)}{b^4}+\frac {3 x^2 \cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {3 \operatorname {CosIntegral}(2 b x)}{b^4}+\frac {3 \log (x)}{b^4}-\frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {6 x \operatorname {CosIntegral}(b x) \sin (b x)}{b^3}+\frac {x^3 \operatorname {CosIntegral}(b x) \sin (b x)}{b}+\frac {13 \sin ^2(b x)}{4 b^4}-\frac {x^2 \sin ^2(b x)}{2 b^2} \] Output:

-1/2*x^2/b^2-3/4*cos(b*x)^2/b^4-6*cos(b*x)*Ci(b*x)/b^4+3*x^2*cos(b*x)*Ci(b 
*x)/b^2+3*Ci(2*b*x)/b^4+3*ln(x)/b^4-2*x*cos(b*x)*sin(b*x)/b^3-6*x*Ci(b*x)* 
sin(b*x)/b^3+x^3*Ci(b*x)*sin(b*x)/b+13/4*sin(b*x)^2/b^4-1/2*x^2*sin(b*x)^2 
/b^2
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.65 \[ \int x^3 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {-3 b^2 x^2-8 \cos (2 b x)+b^2 x^2 \cos (2 b x)+12 \operatorname {CosIntegral}(2 b x)+12 \log (x)+4 \operatorname {CosIntegral}(b x) \left (3 \left (-2+b^2 x^2\right ) \cos (b x)+b x \left (-6+b^2 x^2\right ) \sin (b x)\right )-4 b x \sin (2 b x)}{4 b^4} \] Input:

Integrate[x^3*Cos[b*x]*CosIntegral[b*x],x]
 

Output:

(-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*CosIntegral[b*x]*(3*(-2 + b^2*x^2)*Cos[b*x] + b*x*(-6 + b^2* 
x^2)*Sin[b*x]) - 4*b*x*Sin[2*b*x])/(4*b^4)
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.51, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.750, Rules used = {7068, 27, 3924, 3042, 3791, 15, 7074, 27, 3042, 3791, 15, 7068, 27, 3042, 3044, 15, 7072, 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.

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

\(\Big \downarrow \) 7068

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3924

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7074

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7068

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3044

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7072

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^3*Cos[b*x]*CosIntegral[b*x],x]
 

Output:

(x^3*CosIntegral[b*x]*Sin[b*x])/b - (3*(-((x^2*Cos[b*x]*CosIntegral[b*x])/ 
b) + (x^2/4 + Cos[b*x]^2/(4*b^2) + (x*Cos[b*x]*Sin[b*x])/(2*b))/b + (2*(-( 
(-((Cos[b*x]*CosIntegral[b*x])/b) + (CosIntegral[2*b*x]/2 + Log[x]/2)/b)/b 
) + (x*CosIntegral[b*x]*Sin[b*x])/b - Sin[b*x]^2/(2*b^2)))/b))/b - ((x^2*S 
in[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
 

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 7068

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)* 
(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^m*Sin[a + b*x]*(CosIntegral[c + d* 
x]/b), x] + (-Simp[d/b   Int[(e + f*x)^m*Sin[a + b*x]*(Cos[c + d*x]/(c + d* 
x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Sin[a + b*x]*CosIntegral 
[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

rule 7072

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

rule 7074

Int[CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + 
(b_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*Cos[a + b*x]*(CosIntegral[c + 
 d*x]/b), x] + (Simp[d/b   Int[(e + f*x)^m*Cos[a + b*x]*(Cos[c + d*x]/(c + 
d*x)), x], x] + Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Cos[a + b*x]*CosIntegr 
al[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 6.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x \right ) \left (b^{3} x^{3} \sin \left (b x \right )+3 b^{2} x^{2} \cos \left (b x \right )-6 \cos \left (b x \right )-6 b x \sin \left (b x \right )\right )+\frac {\cos \left (b x \right )^{2} b^{2} x^{2}}{2}-4 b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+b^{2} x^{2}+\sin \left (b x \right )^{2}+3 \ln \left (b x \right )+3 \,\operatorname {Ci}\left (2 b x \right )-3 \cos \left (b x \right )^{2}}{b^{4}}\)	\(116\)
default	\(\frac {\operatorname {Ci}\left (b x \right ) \left (b^{3} x^{3} \sin \left (b x \right )+3 b^{2} x^{2} \cos \left (b x \right )-6 \cos \left (b x \right )-6 b x \sin \left (b x \right )\right )+\frac {\cos \left (b x \right )^{2} b^{2} x^{2}}{2}-4 b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+b^{2} x^{2}+\sin \left (b x \right )^{2}+3 \ln \left (b x \right )+3 \,\operatorname {Ci}\left (2 b x \right )-3 \cos \left (b x \right )^{2}}{b^{4}}\)	\(116\)

Input:

int(x^3*cos(b*x)*Ci(b*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (134) = 268\).

Time = 0.12 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.54 \[ \int x^3 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=-\frac {2 \, \pi b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \cos \left (b x\right ) - 6 \, {\left (\pi ^{3} b^{3} x^{2} - 2 \, \pi ^{3} b\right )} \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + 2 \, {\left (3 \, \pi ^{2} b^{2} x \cos \left (b x\right ) + {\left (\pi ^{2} b^{3} x^{2} - 6 \, \pi ^{2} b + b\right )} \sin \left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, {\left (\pi b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{3} b^{4} x^{3} - 6 \, \pi ^{3} b^{2} x\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (b x\right )}{2 \, \pi ^{3} b^{5}} \] Input:

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

Output:

-1/2*(2*pi*b^2*x*cos(1/2*pi*b^2*x^2)*cos(b*x) - 6*(pi^3*b^3*x^2 - 2*pi^3*b 
)*cos(b*x)*fresnel_cos(b*x) - (6*pi^3*cos(1/2/pi) + (3*pi^2 - 1)*sin(1/2/p 
i))*sqrt(b^2)*fresnel_cos((pi*b*x + 1)*sqrt(b^2)/(pi*b)) - (6*pi^3*cos(1/2 
/pi) + (3*pi^2 - 1)*sin(1/2/pi))*sqrt(b^2)*fresnel_cos((pi*b*x - 1)*sqrt(b 
^2)/(pi*b)) - (6*pi^3*sin(1/2/pi) - (3*pi^2 - 1)*cos(1/2/pi))*sqrt(b^2)*fr 
esnel_sin((pi*b*x + 1)*sqrt(b^2)/(pi*b)) - (6*pi^3*sin(1/2/pi) - (3*pi^2 - 
 1)*cos(1/2/pi))*sqrt(b^2)*fresnel_sin((pi*b*x - 1)*sqrt(b^2)/(pi*b)) + 2* 
(3*pi^2*b^2*x*cos(b*x) + (pi^2*b^3*x^2 - 6*pi^2*b + b)*sin(b*x))*sin(1/2*p 
i*b^2*x^2) - 2*(pi*b*cos(1/2*pi*b^2*x^2) + (pi^3*b^4*x^3 - 6*pi^3*b^2*x)*f 
resnel_cos(b*x))*sin(b*x))/(pi^3*b^5)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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