\(\int x \operatorname {CosIntegral}(b x)^2 \, dx\) [80]

Optimal result

Integrand size = 8, antiderivative size = 75 \[ \int x \operatorname {CosIntegral}(b x)^2 \, dx=-\frac {\cos (b x) \operatorname {CosIntegral}(b x)}{b^2}+\frac {1}{2} x^2 \operatorname {CosIntegral}(b x)^2+\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}+\frac {\log (x)}{2 b^2}-\frac {x \operatorname {CosIntegral}(b x) \sin (b x)}{b}+\frac {\sin ^2(b x)}{2 b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77 \[ \int x \operatorname {CosIntegral}(b x)^2 \, dx=\frac {-\cos (2 b x)+2 b^2 x^2 \operatorname {CosIntegral}(b x)^2+2 \operatorname {CosIntegral}(2 b x)+2 \log (x)-4 \operatorname {CosIntegral}(b x) (\cos (b x)+b x \sin (b x))}{4 b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {7062, 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.

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

\(\Big \downarrow \) 7062

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

\(\Big \downarrow \) 7068

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3044

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7072

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 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 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 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]
 

Maple [A] (verified)

Time = 7.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {b^{2} x^{2} \operatorname {Ci}\left (b x \right )^{2}}{2}-2 \,\operatorname {Ci}\left (b x \right ) \left (\frac {\cos \left (b x \right )}{2}+\frac {b x \sin \left (b x \right )}{2}\right )+\frac {\ln \left (b x \right )}{2}+\frac {\operatorname {Ci}\left (2 b x \right )}{2}-\frac {\cos \left (b x \right )^{2}}{2}}{b^{2}}\)	\(62\)
default	\(\frac {\frac {b^{2} x^{2} \operatorname {Ci}\left (b x \right )^{2}}{2}-2 \,\operatorname {Ci}\left (b x \right ) \left (\frac {\cos \left (b x \right )}{2}+\frac {b x \sin \left (b x \right )}{2}\right )+\frac {\ln \left (b x \right )}{2}+\frac {\operatorname {Ci}\left (2 b x \right )}{2}-\frac {\cos \left (b x \right )^{2}}{2}}{b^{2}}\)	\(62\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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