\(\int x^2 \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx\) [127]

Optimal result

Integrand size = 16, antiderivative size = 173 \[ \int x^2 \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=-\frac {x}{b^2}-\frac {(a-b x) \cos (2 a+2 b x)}{4 b^3}+\frac {2 x \cos (a+b x) \operatorname {CosIntegral}(a+b x)}{b^2}+\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}-\frac {\cos (a+b x) \sin (a+b x)}{b^3}-\frac {2 \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b^3}+\frac {x^2 \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\sin (2 a+2 b x)}{8 b^3}+\frac {\text {Si}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Si}(2 a+2 b x)}{2 b^3} \] Output:

-x/b^2-1/4*(-b*x+a)*cos(2*b*x+2*a)/b^3+2*x*cos(b*x+a)*Ci(b*x+a)/b^2+a*Ci(2 
*b*x+2*a)/b^3+a*ln(b*x+a)/b^3-cos(b*x+a)*sin(b*x+a)/b^3-2*Ci(b*x+a)*sin(b* 
x+a)/b^3+x^2*Ci(b*x+a)*sin(b*x+a)/b-1/8*sin(2*b*x+2*a)/b^3+Si(2*b*x+2*a)/b 
^3-1/2*a^2*Si(2*b*x+2*a)/b^3
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.71 \[ \int x^2 \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {-8 b x-2 a \cos (2 (a+b x))+2 b x \cos (2 (a+b x))+8 a \operatorname {CosIntegral}(2 (a+b x))+8 a \log (a+b x)+8 \operatorname {CosIntegral}(a+b x) \left (2 b x \cos (a+b x)+\left (-2+b^2 x^2\right ) \sin (a+b x)\right )-5 \sin (2 (a+b x))+8 \text {Si}(2 (a+b x))-4 a^2 \text {Si}(2 (a+b x))}{8 b^3} \] Input:

Integrate[x^2*Cos[a + b*x]*CosIntegral[a + b*x],x]
 

Output:

(-8*b*x - 2*a*Cos[2*(a + b*x)] + 2*b*x*Cos[2*(a + b*x)] + 8*a*CosIntegral[ 
2*(a + b*x)] + 8*a*Log[a + b*x] + 8*CosIntegral[a + b*x]*(2*b*x*Cos[a + b* 
x] + (-2 + b^2*x^2)*Sin[a + b*x]) - 5*Sin[2*(a + b*x)] + 8*SinIntegral[2*( 
a + b*x)] - 4*a^2*SinIntegral[2*(a + b*x)])/(8*b^3)
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {7068, 5084, 7074, 7066, 4906, 27, 3042, 3780, 7292, 7293, 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.

\(\displaystyle \int x^2 \operatorname {CosIntegral}(a+b x) \cos (a+b x) \, dx\)

\(\Big \downarrow \) 7068

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

\(\Big \downarrow \) 5084

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

\(\Big \downarrow \) 7074

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {a^2 \text {Si}(2 a+2 b x)}{b^3}-\frac {\sin (2 a+2 b x)}{4 b^3}-\frac {a \cos (2 a+2 b x)}{2 b^3}+\frac {x \cos (2 a+2 b x)}{2 b^2}\right )-\frac {2 \left (-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {a \log (a+b x)}{2 b^2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {\frac {\operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b}}{b}-\frac {x \operatorname {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {x}{2 b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(a+b x) \sin (a+b x)}{b}\)

Input:

Int[x^2*Cos[a + b*x]*CosIntegral[a + b*x],x]
 

Output:

(x^2*CosIntegral[a + b*x]*Sin[a + b*x])/b + (-1/2*(a*Cos[2*a + 2*b*x])/b^3 
 + (x*Cos[2*a + 2*b*x])/(2*b^2) - Sin[2*a + 2*b*x]/(4*b^3) - (a^2*SinInteg 
ral[2*a + 2*b*x])/b^3)/2 - (2*(x/(2*b) - (x*Cos[a + b*x]*CosIntegral[a + b 
*x])/b - (a*CosIntegral[2*a + 2*b*x])/(2*b^2) - (a*Log[a + b*x])/(2*b^2) + 
 (Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + ((CosIntegral[a + b*x]*Sin[a + b*x] 
)/b - SinIntegral[2*a + 2*b*x]/(2*b))/b))/b
 

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 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 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - 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 5084

Int[Cos[w_]^(p_.)*(u_.)*Sin[v_]^(p_.), x_Symbol] :> Simp[1/2^p   Int[u*Sin[ 
2*v]^p, x], x] /; EqQ[w, v] && IntegerQ[p]
 

rule 7066

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

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 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]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 12.62 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x +a \right ) \left (a^{2} \sin \left (b x +a \right )-2 a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Si}\left (2 b x +2 a \right )}{2}-a \cos \left (b x +a \right )^{2}+\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}-\frac {5 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}-\frac {5 b x}{4}-\frac {5 a}{4}+a \ln \left (b x +a \right )+a \,\operatorname {Ci}\left (2 b x +2 a \right )+\operatorname {Si}\left (2 b x +2 a \right )}{b^{3}}\)	\(170\)
default	\(\frac {\operatorname {Ci}\left (b x +a \right ) \left (a^{2} \sin \left (b x +a \right )-2 a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Si}\left (2 b x +2 a \right )}{2}-a \cos \left (b x +a \right )^{2}+\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}-\frac {5 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}-\frac {5 b x}{4}-\frac {5 a}{4}+a \ln \left (b x +a \right )+a \,\operatorname {Ci}\left (2 b x +2 a \right )+\operatorname {Si}\left (2 b x +2 a \right )}{b^{3}}\)	\(170\)

Input:

int(x^2*cos(b*x+a)*Ci(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (168) = 336\).

Time = 0.14 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.39 \[ \int x^2 \cos (a+b x) \operatorname {CosIntegral}(a+b x) \, dx=\frac {4 \, \pi ^{2} b^{2} x \cos \left (b x + a\right ) \operatorname {C}\left (b x + a\right ) - 2 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) \cos \left (b x + a\right ) + 2 \, {\left (\pi ^{2} b^{3} x^{2} - 2 \, \pi ^{2} b\right )} \operatorname {C}\left (b x + a\right ) \sin \left (b x + a\right ) + \sqrt {b^{2}} {\left ({\left (\pi + 2 \, \pi ^{2} a\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (\pi ^{2} {\left (a^{2} - 2\right )} + 2 \, \pi a + 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left ({\left (\pi - 2 \, \pi ^{2} a\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (\pi ^{2} {\left (a^{2} - 2\right )} - 2 \, \pi a + 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left ({\left (\pi ^{2} {\left (a^{2} - 2\right )} + 2 \, \pi a + 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (\pi + 2 \, \pi ^{2} a\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left ({\left (\pi ^{2} {\left (a^{2} - 2\right )} - 2 \, \pi a + 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (\pi - 2 \, \pi ^{2} a\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + \pi a - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - 2 \, {\left (2 \, \pi b \cos \left (b x + a\right ) + {\left (\pi b^{2} x - \pi a b\right )} \sin \left (b x + a\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{2 \, \pi ^{2} b^{4}} \] Input:

integrate(x^2*cos(b*x+a)*fresnel_cos(b*x+a),x, algorithm="fricas")
 

Output:

1/2*(4*pi^2*b^2*x*cos(b*x + a)*fresnel_cos(b*x + a) - 2*b*cos(1/2*pi*b^2*x 
^2 + pi*a*b*x + 1/2*pi*a^2)*cos(b*x + a) + 2*(pi^2*b^3*x^2 - 2*pi^2*b)*fre 
snel_cos(b*x + a)*sin(b*x + a) + sqrt(b^2)*((pi + 2*pi^2*a)*cos(1/2/pi) + 
(pi^2*(a^2 - 2) + 2*pi*a + 1)*sin(1/2/pi))*fresnel_cos((pi*b*x + pi*a + 1) 
*sqrt(b^2)/(pi*b)) - sqrt(b^2)*((pi - 2*pi^2*a)*cos(1/2/pi) + (pi^2*(a^2 - 
 2) - 2*pi*a + 1)*sin(1/2/pi))*fresnel_cos((pi*b*x + pi*a - 1)*sqrt(b^2)/( 
pi*b)) - sqrt(b^2)*((pi^2*(a^2 - 2) + 2*pi*a + 1)*cos(1/2/pi) - (pi + 2*pi 
^2*a)*sin(1/2/pi))*fresnel_sin((pi*b*x + pi*a + 1)*sqrt(b^2)/(pi*b)) + sqr 
t(b^2)*((pi^2*(a^2 - 2) - 2*pi*a + 1)*cos(1/2/pi) - (pi - 2*pi^2*a)*sin(1/ 
2/pi))*fresnel_sin((pi*b*x + pi*a - 1)*sqrt(b^2)/(pi*b)) - 2*(2*pi*b*cos(b 
*x + a) + (pi*b^2*x - pi*a*b)*sin(b*x + a))*sin(1/2*pi*b^2*x^2 + pi*a*b*x 
+ 1/2*pi*a^2))/(pi^2*b^4)
 

Sympy [F]

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

integrate(x**2*cos(b*x+a)*Ci(b*x+a),x)
 

Output:

Integral(x**2*cos(a + b*x)*Ci(a + b*x), x)
 

Maxima [F]

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

integrate(x^2*cos(b*x+a)*fresnel_cos(b*x+a),x, algorithm="maxima")
 

Output:

integrate(x^2*cos(b*x + a)*fresnel_cos(b*x + a), x)
 

Giac [F]

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

integrate(x^2*cos(b*x+a)*fresnel_cos(b*x+a),x, algorithm="giac")
 

Output:

integrate(x^2*cos(b*x + a)*fresnel_cos(b*x + a), x)
 

Mupad [F(-1)]

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

int(x^2*cosint(a + b*x)*cos(a + b*x),x)
 

Output:

int(x^2*cosint(a + b*x)*cos(a + b*x), x)
 

Reduce [F]

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

int(x^2*cos(b*x+a)*Ci(b*x+a),x)
 

Output:

int(ci(a + b*x)*cos(a + b*x)*x**2,x)