\(\int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx\) [132]

Optimal result

Integrand size = 13, antiderivative size = 154 \[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\cos (a+b x) \operatorname {CosIntegral}(c+d x)}{b}+\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94 \[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\frac {-4 \cos (a+b x) \operatorname {CosIntegral}(c+d x)+\left (\operatorname {ExpIntegralEi}\left (-\frac {i (b-d) (c+d x)}{d}\right )+\operatorname {ExpIntegralEi}\left (-\frac {i (b+d) (c+d x)}{d}\right )\right ) \left (\cos \left (a-\frac {b c}{d}\right )-i \sin \left (a-\frac {b c}{d}\right )\right )+\left (\operatorname {ExpIntegralEi}\left (\frac {i (b-d) (c+d x)}{d}\right )+\operatorname {ExpIntegralEi}\left (\frac {i (b+d) (c+d x)}{d}\right )\right ) \left (\cos \left (a-\frac {b c}{d}\right )+i \sin \left (a-\frac {b c}{d}\right )\right )}{4 b} \] Input:

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

Output:

(-4*Cos[a + b*x]*CosIntegral[c + d*x] + (ExpIntegralEi[((-I)*(b - d)*(c + 
d*x))/d] + ExpIntegralEi[((-I)*(b + d)*(c + d*x))/d])*(Cos[a - (b*c)/d] - 
I*Sin[a - (b*c)/d]) + (ExpIntegralEi[(I*(b - d)*(c + d*x))/d] + ExpIntegra 
lEi[(I*(b + d)*(c + d*x))/d])*(Cos[a - (b*c)/d] + I*Sin[a - (b*c)/d]))/(4* 
b)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7072, 4929, 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.

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

\(\Big \downarrow \) 7072

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

\(\Big \downarrow \) 4929

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((Cos[a + b*x]*CosIntegral[c + d*x])/b) + (d*((Cos[a - (b*c)/d]*CosIntegr 
al[(c*(b - d))/d + (b - d)*x])/(2*d) + (Cos[a - (b*c)/d]*CosIntegral[(c*(b 
 + d))/d + (b + d)*x])/(2*d) - (Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))/d 
 + (b - d)*x])/(2*d) - (Sin[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + 
d)*x])/(2*d)))/b
 

Defintions of rubi rules used

rule 2009

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

rule 4929

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

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 = 0.98 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.71

\[\frac {-\frac {\operatorname {Ci}\left (d x +c \right ) d \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{b}+\frac {d \left (\frac {d \left (\frac {\operatorname {Si}\left (\left (-1+\frac {b}{d}\right ) \left (d x +c \right )+a -\frac {c b}{d}+\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\left (-1+\frac {b}{d}\right ) \left (d x +c \right )+a -\frac {c b}{d}+\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}\right )}{2}+\frac {d \left (\frac {\operatorname {Si}\left (\left (1+\frac {b}{d}\right ) \left (d x +c \right )+a -\frac {c b}{d}+\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\left (1+\frac {b}{d}\right ) \left (d x +c \right )+a -\frac {c b}{d}+\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}\right )}{2}\right )}{b}}{d}\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.55 \[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=-\frac {2 \, d \cos \left (b x + a\right ) \operatorname {C}\left (d x + c\right ) - \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) + \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )}{2 \, b d} \] Input:

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

Output:

-1/2*(2*d*cos(b*x + a)*fresnel_cos(d*x + c) - sqrt(d^2)*cos(a - b*c/d - 1/ 
2*b^2/(pi*d^2))*fresnel_cos((pi*d^2*x + pi*c*d + b)*sqrt(d^2)/(pi*d^2)) - 
sqrt(d^2)*cos(a - b*c/d + 1/2*b^2/(pi*d^2))*fresnel_cos((pi*d^2*x + pi*c*d 
 - b)*sqrt(d^2)/(pi*d^2)) - sqrt(d^2)*fresnel_sin((pi*d^2*x + pi*c*d - b)* 
sqrt(d^2)/(pi*d^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))*sin(a - b*c/d - 1/2*b^2/(p 
i*d^2)))/(b*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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