\(\int \cos (a+b x) \operatorname {CosIntegral}(c+d x) \, dx\) [135]

Optimal result

Integrand size = 13, antiderivative size = 153 \[ \int \cos (a+b x) \operatorname {CosIntegral}(c+d x) \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b}-\frac {\operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b}+\frac {\operatorname {CosIntegral}(c+d x) \sin (a+b x)}{b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.90 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00 \[ \int \cos (a+b x) \operatorname {CosIntegral}(c+d x) \, dx=\frac {i e^{-\frac {i (b c+a d)}{d}} \left (-e^{\frac {2 i b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {i (b-d) (c+d x)}{d}\right )+e^{2 i a} \operatorname {ExpIntegralEi}\left (\frac {i (b-d) (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {i (b+d) (c+d x)}{d}\right )+e^{2 i a} \operatorname {ExpIntegralEi}\left (\frac {i (b+d) (c+d x)}{d}\right )\right )+4 \operatorname {CosIntegral}(c+d x) \sin (a+b x)}{4 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {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.

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

\(\Big \downarrow \) 7066

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

\(\Big \downarrow \) 4930

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(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) + (Cos[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
 

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 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]
 

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.74

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.56 \[ \int \cos (a+b x) \operatorname {CosIntegral}(c+d x) \, dx=\frac {2 \, d \operatorname {C}\left (d x + c\right ) \sin \left (b x + a\right ) - \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) + \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) - \sqrt {d^{2}} \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )}{2 \, b d} \] Input:

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

Output:

1/2*(2*d*fresnel_cos(d*x + c)*sin(b*x + a) - sqrt(d^2)*cos(a - b*c/d - 1/2 
*b^2/(pi*d^2))*fresnel_sin((pi*d^2*x + pi*c*d + b)*sqrt(d^2)/(pi*d^2)) + s 
qrt(d^2)*cos(a - b*c/d + 1/2*b^2/(pi*d^2))*fresnel_sin((pi*d^2*x + pi*c*d 
- b)*sqrt(d^2)/(pi*d^2)) - sqrt(d^2)*fresnel_cos((pi*d^2*x + pi*c*d - b)*s 
qrt(d^2)/(pi*d^2))*sin(a - b*c/d + 1/2*b^2/(pi*d^2)) - sqrt(d^2)*fresnel_c 
os((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)))/(b*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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