Integrand size = 13, antiderivative size = 154 \[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\cos (a+b x) \operatorname {CosIntegral}(c+d x)}{b}+\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b} \]
1/2*Ci(c*(b-d)/d+(b-d)*x)*cos(a-b*c/d)/b+1/2*Ci(c*(b+d)/d+(b+d)*x)*cos(a-b *c/d)/b-Ci(d*x+c)*cos(b*x+a)/b-1/2*Si(c*(b-d)/d+(b-d)*x)*sin(a-b*c/d)/b-1/ 2*Si(c*(b+d)/d+(b+d)*x)*sin(a-b*c/d)/b
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94 \[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\frac {-4 \cos (a+b x) \operatorname {CosIntegral}(c+d x)+\left (\operatorname {ExpIntegralEi}\left (-\frac {i (b-d) (c+d x)}{d}\right )+\operatorname {ExpIntegralEi}\left (-\frac {i (b+d) (c+d x)}{d}\right )\right ) \left (\cos \left (a-\frac {b c}{d}\right )-i \sin \left (a-\frac {b c}{d}\right )\right )+\left (\operatorname {ExpIntegralEi}\left (\frac {i (b-d) (c+d x)}{d}\right )+\operatorname {ExpIntegralEi}\left (\frac {i (b+d) (c+d x)}{d}\right )\right ) \left (\cos \left (a-\frac {b c}{d}\right )+i \sin \left (a-\frac {b c}{d}\right )\right )}{4 b} \]
(-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)
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 |
\(\displaystyle \frac {d \left (\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}\right )}{b}-\frac {\cos (a+b x) \operatorname {CosIntegral}(c+d x)}{b}\) |
-((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
3.2.32.3.1 Defintions of rubi rules used
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]
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]
Time = 1.50 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.77
method | result | size |
default | \(\frac {-\frac {\operatorname {Ci}\left (d x +c \right ) d \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{b}+\frac {d \left (\frac {d \left (-\frac {\operatorname {Si}\left (-\left (-1+\frac {b}{d}\right ) \left (d x +c \right )-a +\frac {b c}{d}-\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\left (-1+\frac {b}{d}\right ) \left (d x +c \right )+a -\frac {b c}{d}+\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}\right )}{2}+\frac {d \left (-\frac {\operatorname {Si}\left (-\left (1+\frac {b}{d}\right ) \left (d x +c \right )-a +\frac {b c}{d}-\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\left (1+\frac {b}{d}\right ) \left (d x +c \right )+a -\frac {b c}{d}+\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}\right )}{2}\right )}{b}}{d}\) | \(272\) |
(-Ci(d*x+c)/b*d*cos(1/d*b*(d*x+c)+(a*d-b*c)/d)+1/b*d*(1/2*d*(-Si(-(-1+b/d) *(d*x+c)-a+b*c/d-(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d+Ci((-1+b/d)*(d*x+c)+a-b *c/d+(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d)+1/2*d*(-Si(-(1+b/d)*(d*x+c)-a+b*c/ d-(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d+Ci((1+b/d)*(d*x+c)+a-b*c/d+(-a*d+b*c)/ d)*cos((-a*d+b*c)/d)/d)))/d
Time = 0.26 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.55 \[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=-\frac {2 \, d \cos \left (b x + a\right ) \operatorname {C}\left (d x + c\right ) - \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - \sqrt {d^{2}} \cos \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) \operatorname {C}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) - \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d - b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} + \frac {b^{2}}{2 \, \pi d^{2}}\right ) + \sqrt {d^{2}} \operatorname {S}\left (\frac {{\left (\pi d^{2} x + \pi c d + b\right )} \sqrt {d^{2}}}{\pi d^{2}}\right ) \sin \left (a - \frac {b c}{d} - \frac {b^{2}}{2 \, \pi d^{2}}\right )}{2 \, b d} \]
-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)
\[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\int \sin {\left (a + b x \right )} \operatorname {Ci}{\left (c + d x \right )}\, dx \]
\[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\int { \operatorname {C}\left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \]
\[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\int { \operatorname {C}\left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \]
Timed out. \[ \int \operatorname {CosIntegral}(c+d x) \sin (a+b x) \, dx=\int \mathrm {cosint}\left (c+d\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]