3.1.79 \(\int x^2 \operatorname {CosIntegral}(b x)^2 \, dx\) [79]

3.1.79.1 Optimal result

Integrand size = 10, antiderivative size = 112 \[ \int x^2 \operatorname {CosIntegral}(b x)^2 \, dx=\frac {x}{2 b^2}-\frac {4 x \cos (b x) \operatorname {CosIntegral}(b x)}{3 b^2}+\frac {1}{3} x^3 \operatorname {CosIntegral}(b x)^2+\frac {5 \cos (b x) \sin (b x)}{6 b^3}+\frac {4 \operatorname {CosIntegral}(b x) \sin (b x)}{3 b^3}-\frac {2 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{3 b}+\frac {x \sin ^2(b x)}{3 b^2}-\frac {2 \text {Si}(2 b x)}{3 b^3} \]

output

1/2*x/b^2+1/3*x^3*Ci(b*x)^2-4/3*x*Ci(b*x)*cos(b*x)/b^2-2/3*Si(2*b*x)/b^3+4 
/3*Ci(b*x)*sin(b*x)/b^3-2/3*x^2*Ci(b*x)*sin(b*x)/b+5/6*cos(b*x)*sin(b*x)/b 
^3+1/3*x*sin(b*x)^2/b^2
 

3.1.79.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.70 \[ \int x^2 \operatorname {CosIntegral}(b x)^2 \, dx=\frac {8 b x-2 b x \cos (2 b x)+4 b^3 x^3 \operatorname {CosIntegral}(b x)^2-8 \operatorname {CosIntegral}(b x) \left (2 b x \cos (b x)+\left (-2+b^2 x^2\right ) \sin (b x)\right )+5 \sin (2 b x)-8 \text {Si}(2 b x)}{12 b^3} \]

input

Integrate[x^2*CosIntegral[b*x]^2,x]
 

output

(8*b*x - 2*b*x*Cos[2*b*x] + 4*b^3*x^3*CosIntegral[b*x]^2 - 8*CosIntegral[b 
*x]*(2*b*x*Cos[b*x] + (-2 + b^2*x^2)*Sin[b*x]) + 5*Sin[2*b*x] - 8*SinInteg 
ral[2*b*x])/(12*b^3)
 

3.1.79.3 Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.39, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.800, Rules used = {7062, 7068, 27, 3924, 3042, 3115, 24, 7074, 27, 3042, 3115, 24, 7066, 27, 4906, 27, 3042, 3780}

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.

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

\(\Big \downarrow \) 7062

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

\(\Big \downarrow \) 7068

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3924

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7074

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

input

Int[x^2*CosIntegral[b*x]^2,x]
 

output

(x^3*CosIntegral[b*x]^2)/3 - (2*((x^2*CosIntegral[b*x]*Sin[b*x])/b - ((x*S 
in[b*x]^2)/(2*b) - (x/2 - (Cos[b*x]*Sin[b*x])/(2*b))/(2*b))/b - (2*(-((x*C 
os[b*x]*CosIntegral[b*x])/b) + (x/2 + (Cos[b*x]*Sin[b*x])/(2*b))/b + ((Cos 
Integral[b*x]*Sin[b*x])/b - SinIntegral[2*b*x]/(2*b))/b))/b))/3
 

3.1.79.3.1 Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

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 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 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 7062

Int[CosIntegral[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(CosI 
ntegral[b*x]^2/(m + 1)), x] - Simp[2/(m + 1)   Int[x^m*Cos[b*x]*CosIntegral 
[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
 

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]
 

3.1.79.4 Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {\frac {b^{3} x^{3} \operatorname {Ci}\left (b x \right )^{2}}{3}-2 \,\operatorname {Ci}\left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (b x \right )}{3}-\frac {2 \sin \left (b x \right )}{3}+\frac {2 b x \cos \left (b x \right )}{3}\right )-\frac {b x \cos \left (b x \right )^{2}}{3}+\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{6}+\frac {5 b x}{6}-\frac {2 \,\operatorname {Si}\left (2 b x \right )}{3}}{b^{3}}\)	\(84\)
default	\(\frac {\frac {b^{3} x^{3} \operatorname {Ci}\left (b x \right )^{2}}{3}-2 \,\operatorname {Ci}\left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (b x \right )}{3}-\frac {2 \sin \left (b x \right )}{3}+\frac {2 b x \cos \left (b x \right )}{3}\right )-\frac {b x \cos \left (b x \right )^{2}}{3}+\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{6}+\frac {5 b x}{6}-\frac {2 \,\operatorname {Si}\left (2 b x \right )}{3}}{b^{3}}\)	\(84\)

input

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

output

1/b^3*(1/3*b^3*x^3*Ci(b*x)^2-2*Ci(b*x)*(1/3*b^2*x^2*sin(b*x)-2/3*sin(b*x)+ 
2/3*b*x*cos(b*x))-1/3*b*x*cos(b*x)^2+5/6*sin(b*x)*cos(b*x)+5/6*b*x-2/3*Si( 
2*b*x))
 

3.1.79.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.99 \[ \int x^2 \operatorname {CosIntegral}(b x)^2 \, dx=\frac {4 \, \pi ^{2} b^{4} x^{3} \operatorname {C}\left (b x\right )^{2} - 8 \, \pi b^{3} x^{2} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 10 \, b^{2} x - 16 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + 5 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi ^{2} b^{4}} \]

input

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

output

1/12*(4*pi^2*b^4*x^3*fresnel_cos(b*x)^2 - 8*pi*b^3*x^2*fresnel_cos(b*x)*si 
n(1/2*pi*b^2*x^2) - 4*b^2*x*cos(1/2*pi*b^2*x^2)^2 + 10*b^2*x - 16*b*cos(1/ 
2*pi*b^2*x^2)*fresnel_cos(b*x) + 5*sqrt(2)*sqrt(b^2)*fresnel_cos(sqrt(2)*s 
qrt(b^2)*x))/(pi^2*b^4)
 

3.1.79.6 Sympy [F]

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

input

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

output

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

3.1.79.7 Maxima [F]

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

input

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

output

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

3.1.79.8 Giac [F]

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

input

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

output

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

3.1.79.9 Mupad [F(-1)]

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

input

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

output

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