\(\int x \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx\) [131]

Optimal result

Integrand size = 14, antiderivative size = 371 \[ \int x \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=-\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}-\frac {x \cos (a+b x) \operatorname {CosIntegral}(c+d x)}{b}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}-\frac {\operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}+\frac {\operatorname {CosIntegral}(c+d x) \sin (a+b x)}{b^2}+\frac {\sin (a-c+(b-d) x)}{2 b (b-d)}+\frac {\sin (a+c+(b+d) x)}{2 b (b+d)}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.68 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.86 \[ \int x \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=-\frac {\frac {e^{-i a} \left (-i b d e^{-i c} \left (\frac {e^{-i (b+d) x}}{b+d}+\frac {e^{i (2 c-b x+d x)}}{b-d}\right )+(b c+i d) e^{\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {i (b-d) (c+d x)}{d}\right )+(b c+i d) e^{\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {i (b+d) (c+d x)}{d}\right )\right )}{d}+\frac {e^{i a} \left (i b d e^{-i c} \left (\frac {e^{i (b-d) x}}{b-d}+\frac {e^{i (2 c+(b+d) x)}}{b+d}\right )+(b c-i d) e^{-\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {i (b-d) (c+d x)}{d}\right )+(b c-i d) e^{-\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {i (b+d) (c+d x)}{d}\right )\right )}{d}+4 \operatorname {CosIntegral}(c+d x) (b x \cos (a+b x)-\sin (a+b x))}{4 b^2} \] Input:

Integrate[x*CosIntegral[c + d*x]*Sin[a + b*x],x]
 

Output:

-1/4*((((-I)*b*d*(1/((b + d)*E^(I*(b + d)*x)) + E^(I*(2*c - b*x + d*x))/(b 
 - d)))/E^(I*c) + (b*c + I*d)*E^((I*b*c)/d)*ExpIntegralEi[((-I)*(b - d)*(c 
 + d*x))/d] + (b*c + I*d)*E^((I*b*c)/d)*ExpIntegralEi[((-I)*(b + d)*(c + d 
*x))/d])/(d*E^(I*a)) + (E^(I*a)*((I*b*d*(E^(I*(b - d)*x)/(b - d) + E^(I*(2 
*c + (b + d)*x))/(b + d)))/E^(I*c) + ((b*c - I*d)*ExpIntegralEi[(I*(b - d) 
*(c + d*x))/d])/E^((I*b*c)/d) + ((b*c - I*d)*ExpIntegralEi[(I*(b + d)*(c + 
 d*x))/d])/E^((I*b*c)/d)))/d + 4*CosIntegral[c + d*x]*(b*x*Cos[a + b*x] - 
Sin[a + b*x]))/b^2
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {7074, 5120, 2009, 7066, 4930, 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.

\(\displaystyle \int x \sin (a+b x) \operatorname {CosIntegral}(c+d x) \, dx\)

\(\Big \downarrow \) 7074

\(\displaystyle \frac {\int \cos (a+b x) \operatorname {CosIntegral}(c+d x)dx}{b}+\frac {d \int \frac {x \cos (a+b x) \cos (c+d x)}{c+d x}dx}{b}-\frac {x \cos (a+b x) \operatorname {CosIntegral}(c+d x)}{b}\)

\(\Big \downarrow \) 5120

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 4930

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

\(\Big \downarrow \) 2009

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

Input:

Int[x*CosIntegral[c + d*x]*Sin[a + b*x],x]
 

Output:

-((x*Cos[a + b*x]*CosIntegral[c + d*x])/b) + (d*(-1/2*(c*Cos[a - (b*c)/d]* 
CosIntegral[(c*(b - d))/d + (b - d)*x])/d^2 - (c*Cos[a - (b*c)/d]*CosInteg 
ral[(c*(b + d))/d + (b + d)*x])/(2*d^2) + Sin[a - c + (b - d)*x]/(2*(b - d 
)*d) + Sin[a + c + (b + d)*x]/(2*d*(b + d)) + (c*Sin[a - (b*c)/d]*SinInteg 
ral[(c*(b - d))/d + (b - d)*x])/(2*d^2) + (c*Sin[a - (b*c)/d]*SinIntegral[ 
(c*(b + d))/d + (b + d)*x])/(2*d^2)))/b + ((CosIntegral[c + d*x]*Sin[a + b 
*x])/b - (d*((CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2* 
d) + (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*d) + (Co 
s[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*d) + (Cos[a - (b 
*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*d)))/b)/b
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4930

Int[Cos[(c_.) + (d_.)*(x_)]^(q_.)*((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x 
]^p*Cos[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IGtQ[q, 0]
 

rule 5120

Int[Cos[(a_.) + (b_.)*(x_)]^(m_.)*Cos[(c_.) + (d_.)*(x_)]^(n_.)*(u_.), x_Sy 
mbol] :> Int[ExpandTrigReduce[u, Cos[a + b*x]^m*Cos[c + d*x]^n, x], x] /; F 
reeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1207\) vs. \(2(351)=702\).

Time = 1.48 (sec) , antiderivative size = 1208, normalized size of antiderivative = 3.26

\[\text {Expression too large to display}\]

Input:

int(x*Ci(d*x+c)*sin(b*x+a),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.22 \[ \int x \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=-\frac {2 \, \pi b d^{3} x \cos \left (b x + a\right ) \operatorname {C}\left (d x + c\right ) - 2 \, \pi d^{3} \operatorname {C}\left (d x + c\right ) \sin \left (b x + a\right ) - 2 \, b d^{2} \cos \left (b x + a\right ) \sin \left (\frac {1}{2} \, \pi d^{2} x^{2} + \pi c d x + \frac {1}{2} \, \pi c^{2}\right ) + {\left (\pi d^{2} \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) + {\left (\pi b c d + b^{2}\right )} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) + {\left (\pi d^{2} \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) + {\left (\pi b c d - b^{2}\right )} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) + {\left (\pi d^{2} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) - {\left (\pi b c d + b^{2}\right )} \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - {\left (\pi d^{2} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) - {\left (\pi b c d - b^{2}\right )} \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right )\right )} \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right )}{2 \, \pi b^{2} d^{3}} \] Input:

integrate(x*fresnel_cos(d*x+c)*sin(b*x+a),x, algorithm="fricas")
 

Output:

-1/2*(2*pi*b*d^3*x*cos(b*x + a)*fresnel_cos(d*x + c) - 2*pi*d^3*fresnel_co 
s(d*x + c)*sin(b*x + a) - 2*b*d^2*cos(b*x + a)*sin(1/2*pi*d^2*x^2 + pi*c*d 
*x + 1/2*pi*c^2) + (pi*d^2*sin(a - b*c/d - 1/2*b^2/(pi*d^2)) + (pi*b*c*d + 
 b^2)*cos(a - b*c/d - 1/2*b^2/(pi*d^2)))*sqrt(d^2)*fresnel_cos((pi*d^2*x + 
 pi*c*d + b)*sqrt(d^2)/(pi*d^2)) + (pi*d^2*sin(a - b*c/d + 1/2*b^2/(pi*d^2 
)) + (pi*b*c*d - b^2)*cos(a - b*c/d + 1/2*b^2/(pi*d^2)))*sqrt(d^2)*fresnel 
_cos((pi*d^2*x + pi*c*d - b)*sqrt(d^2)/(pi*d^2)) + (pi*d^2*cos(a - b*c/d - 
 1/2*b^2/(pi*d^2)) - (pi*b*c*d + b^2)*sin(a - b*c/d - 1/2*b^2/(pi*d^2)))*s 
qrt(d^2)*fresnel_sin((pi*d^2*x + pi*c*d + b)*sqrt(d^2)/(pi*d^2)) - (pi*d^2 
*cos(a - b*c/d + 1/2*b^2/(pi*d^2)) - (pi*b*c*d - b^2)*sin(a - b*c/d + 1/2* 
b^2/(pi*d^2)))*sqrt(d^2)*fresnel_sin((pi*d^2*x + pi*c*d - b)*sqrt(d^2)/(pi 
*d^2)))/(pi*b^2*d^3)
 

Sympy [F]

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

integrate(x*Ci(d*x+c)*sin(b*x+a),x)
 

Output:

Integral(x*sin(a + b*x)*Ci(c + d*x), x)
 

Maxima [F]

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

integrate(x*fresnel_cos(d*x+c)*sin(b*x+a),x, algorithm="maxima")
 

Output:

integrate(x*fresnel_cos(d*x + c)*sin(b*x + a), x)
 

Giac [F]

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

integrate(x*fresnel_cos(d*x+c)*sin(b*x+a),x, algorithm="giac")
 

Output:

integrate(x*fresnel_cos(d*x + c)*sin(b*x + a), x)
 

Mupad [F(-1)]

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

int(x*cosint(c + d*x)*sin(a + b*x),x)
 

Output:

int(x*cosint(c + d*x)*sin(a + b*x), x)
 

Reduce [F]

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

int(x*Ci(d*x+c)*sin(b*x+a),x)
                                                                                    
                                                                                    
 

Output:

int(ci(c + d*x)*sin(a + b*x)*x,x)