\(\int x \operatorname {CosIntegral}(b x) \sin (b x) \, dx\) [111]

Optimal result

Integrand size = 10, antiderivative size = 62 \[ \int x \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\frac {x}{2 b}-\frac {x \cos (b x) \operatorname {CosIntegral}(b x)}{b}+\frac {\cos (b x) \sin (b x)}{2 b^2}+\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {\text {Si}(2 b x)}{2 b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int x \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\frac {2 b x+\operatorname {CosIntegral}(b x) (-4 b x \cos (b x)+4 \sin (b x))+\sin (2 b x)-2 \text {Si}(2 b x)}{4 b^2} \] Input:

Integrate[x*CosIntegral[b*x]*Sin[b*x],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {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.

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

\(\Big \downarrow \) 7074

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

\(\displaystyle \frac {\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\text {Si}(2 b x)}{2 b}}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\)

Input:

Int[x*CosIntegral[b*x]*Sin[b*x],x]
 

Output:

-((x*Cos[b*x]*CosIntegral[b*x])/b) + (x/2 + (Cos[b*x]*Sin[b*x])/(2*b))/b + 
 ((CosIntegral[b*x]*Sin[b*x])/b - SinIntegral[2*b*x]/(2*b))/b
 

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 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 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 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 = 4.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x \right ) \left (\sin \left (b x \right )-b x \cos \left (b x \right )\right )+\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}-\frac {\operatorname {Si}\left (2 b x \right )}{2}}{b^{2}}\)	\(45\)
default	\(\frac {\operatorname {Ci}\left (b x \right ) \left (\sin \left (b x \right )-b x \cos \left (b x \right )\right )+\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}-\frac {\operatorname {Si}\left (2 b x \right )}{2}}{b^{2}}\)	\(45\)

Input:

int(x*Ci(b*x)*sin(b*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (56) = 112\).

Time = 0.12 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.53 \[ \int x \operatorname {CosIntegral}(b x) \sin (b x) \, dx=-\frac {2 \, \pi b^{2} x \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - 2 \, \pi b \operatorname {C}\left (b x\right ) \sin \left (b x\right ) - 2 \, b \cos \left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - \sqrt {b^{2}} {\left (\pi \sin \left (\frac {1}{2 \, \pi }\right ) - \cos \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left (\pi \sin \left (\frac {1}{2 \, \pi }\right ) - \cos \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) + \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right )}{2 \, \pi b^{3}} \] Input:

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

Output:

-1/2*(2*pi*b^2*x*cos(b*x)*fresnel_cos(b*x) - 2*pi*b*fresnel_cos(b*x)*sin(b 
*x) - 2*b*cos(b*x)*sin(1/2*pi*b^2*x^2) - sqrt(b^2)*(pi*sin(1/2/pi) - cos(1 
/2/pi))*fresnel_cos((pi*b*x + 1)*sqrt(b^2)/(pi*b)) + sqrt(b^2)*(pi*sin(1/2 
/pi) - cos(1/2/pi))*fresnel_cos((pi*b*x - 1)*sqrt(b^2)/(pi*b)) + sqrt(b^2) 
*(pi*cos(1/2/pi) + sin(1/2/pi))*fresnel_sin((pi*b*x + 1)*sqrt(b^2)/(pi*b)) 
 - sqrt(b^2)*(pi*cos(1/2/pi) + sin(1/2/pi))*fresnel_sin((pi*b*x - 1)*sqrt( 
b^2)/(pi*b)))/(pi*b^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x*fresnel_cos(b*x)*sin(b*x), x)
 

Giac [F]

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

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

Output:

integrate(x*fresnel_cos(b*x)*sin(b*x), x)
 

Mupad [F(-1)]

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

int(x*cosint(b*x)*sin(b*x),x)
 

Output:

int(x*cosint(b*x)*sin(b*x), x)
 

Reduce [F]

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

int(x*Ci(b*x)*sin(b*x),x)
 

Output:

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