Integrand size = 14, antiderivative size = 371 \[ \int x \sin (a+b x) \text {Si}(c+d x) \, dx=\frac {\cos (a-c+(b-d) x)}{2 b (b-d)}-\frac {\cos (a+c+(b+d) x)}{2 b (b+d)}-\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b d}-\frac {c \operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b d}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}+\frac {\sin (a+b x) \text {Si}(c+d x)}{b^2}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2} \]
-1/2*Ci(c*(b-d)/d+(b-d)*x)*cos(a-b*c/d)/b^2+1/2*Ci(c*(b+d)/d+(b+d)*x)*cos( a-b*c/d)/b^2+1/2*cos(a-c+(b-d)*x)/b/(b-d)-1/2*cos(a+c+(b+d)*x)/b/(b+d)+1/2 *c*cos(a-b*c/d)*Si(c*(b-d)/d+(b-d)*x)/b/d-x*cos(b*x+a)*Si(d*x+c)/b-1/2*c*c os(a-b*c/d)*Si(c*(b+d)/d+(b+d)*x)/b/d+1/2*c*Ci(c*(b-d)/d+(b-d)*x)*sin(a-b* c/d)/b/d-1/2*c*Ci(c*(b+d)/d+(b+d)*x)*sin(a-b*c/d)/b/d+1/2*Si(c*(b-d)/d+(b- d)*x)*sin(a-b*c/d)/b^2-1/2*Si(c*(b+d)/d+(b+d)*x)*sin(a-b*c/d)/b^2+Si(d*x+c )*sin(b*x+a)/b^2
Result contains complex when optimal does not.
Time = 3.59 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.10 \[ \int x \sin (a+b x) \text {Si}(c+d x) \, dx=\frac {e^{-i a} \left (-i (b c-i d) e^{2 i a-\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {i (b-d) (c+d x)}{d}\right )+\frac {e^{-\frac {i (b+d) (c+d x)}{d}} \left (b d e^{\frac {i b c}{d}} \left (b \left (-1+e^{2 i (a+b x)}\right )+d \left (1+e^{2 i (a+b x)}\right )\right )+(-i b c+d) \left (b^2-d^2\right ) e^{i \left (c+\frac {2 b c}{d}+(b+d) x\right )} \operatorname {ExpIntegralEi}\left (-\frac {i (b+d) (c+d x)}{d}\right )\right )}{(b-d) (b+d)}\right )}{4 b^2 d}+\frac {e^{-i a} \left (-\left ((-i b c+d) e^{\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {i (b-d) (c+d x)}{d}\right )\right )+\frac {e^{-\frac {i b c}{d}} \left (b d e^{\frac {i (b (c-d x)+d (c+d x))}{d}} \left (b+d-b e^{2 i (a+b x)}+d e^{2 i (a+b x)}\right )+(i b c+d) \left (b^2-d^2\right ) e^{2 i a} \operatorname {ExpIntegralEi}\left (\frac {i (b+d) (c+d x)}{d}\right )\right )}{(b-d) (b+d)}\right )}{4 b^2 d}-\frac {(b x \cos (a+b x)-\sin (a+b x)) \text {Si}(c+d x)}{b^2} \]
((-I)*(b*c - I*d)*E^((2*I)*a - (I*b*c)/d)*ExpIntegralEi[(I*(b - d)*(c + d* x))/d] + (b*d*E^((I*b*c)/d)*(b*(-1 + E^((2*I)*(a + b*x))) + d*(1 + E^((2*I )*(a + b*x)))) + ((-I)*b*c + d)*(b^2 - d^2)*E^(I*(c + (2*b*c)/d + (b + d)* x))*ExpIntegralEi[((-I)*(b + d)*(c + d*x))/d])/((b - d)*(b + d)*E^((I*(b + d)*(c + d*x))/d)))/(4*b^2*d*E^(I*a)) + (-(((-I)*b*c + d)*E^((I*b*c)/d)*Ex pIntegralEi[((-I)*(b - d)*(c + d*x))/d]) + (b*d*E^((I*(b*(c - d*x) + d*(c + d*x)))/d)*(b + d - b*E^((2*I)*(a + b*x)) + d*E^((2*I)*(a + b*x))) + (I*b *c + d)*(b^2 - d^2)*E^((2*I)*a)*ExpIntegralEi[(I*(b + d)*(c + d*x))/d])/(( b - d)*(b + d)*E^((I*b*c)/d)))/(4*b^2*d*E^(I*a)) - ((b*x*Cos[a + b*x] - Si n[a + b*x])*SinIntegral[c + d*x])/b^2
Time = 1.55 (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 = {7067, 7071, 4928, 2009, 7293, 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.
| \(\displaystyle \int x \sin (a+b x) \text {Si}(c+d x) \, dx\) |
| \(\Big \downarrow \) 7067 |
| \(\displaystyle \frac {\int \cos (a+b x) \text {Si}(c+d x)dx}{b}+\frac {d \int \frac {x \cos (a+b x) \sin (c+d x)}{c+d x}dx}{b}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 7071 |
| \(\displaystyle \frac {\frac {\sin (a+b x) \text {Si}(c+d x)}{b}-\frac {d \int \frac {\sin (a+b x) \sin (c+d x)}{c+d x}dx}{b}}{b}+\frac {d \int \frac {x \cos (a+b x) \sin (c+d x)}{c+d x}dx}{b}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 4928 |
| \(\displaystyle \frac {\frac {\sin (a+b x) \text {Si}(c+d x)}{b}-\frac {d \int \left (\frac {\cos (a-c+(b-d) x)}{2 (c+d x)}-\frac {\cos (a+c+(b+d) x)}{2 (c+d x)}\right )dx}{b}}{b}+\frac {d \int \frac {x \cos (a+b x) \sin (c+d x)}{c+d x}dx}{b}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \int \frac {x \cos (a+b x) \sin (c+d x)}{c+d x}dx}{b}+\frac {\frac {\sin (a+b x) \text {Si}(c+d x)}{b}-\frac {d \left (\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}\right )}{b}}{b}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {d \int \left (\frac {\cos (a+b x) \sin (c+d x)}{d}-\frac {c \cos (a+b x) \sin (c+d x)}{d (c+d x)}\right )dx}{b}+\frac {\frac {\sin (a+b x) \text {Si}(c+d x)}{b}-\frac {d \left (\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}\right )}{b}}{b}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (\frac {c \sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {\cos (a+x (b-d)-c)}{2 d (b-d)}-\frac {\cos (a+x (b+d)+c)}{2 d (b+d)}\right )}{b}+\frac {\frac {\sin (a+b x) \text {Si}(c+d x)}{b}-\frac {d \left (\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}\right )}{b}}{b}-\frac {x \cos (a+b x) \text {Si}(c+d x)}{b}\) |
-((x*Cos[a + b*x]*SinIntegral[c + d*x])/b) + (d*(Cos[a - c + (b - d)*x]/(2 *(b - d)*d) - Cos[a + c + (b + d)*x]/(2*d*(b + d)) + (c*CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*d^2) - (c*CosIntegral[(c*(b + d) )/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*d^2) + (c*Cos[a - (b*c)/d]*SinIntegr al[(c*(b - d))/d + (b - d)*x])/(2*d^2) - (c*Cos[a - (b*c)/d]*SinIntegral[( c*(b + d))/d + (b + d)*x])/(2*d^2)))/b + ((Sin[a + b*x]*SinIntegral[c + d* x])/b - (d*((Cos[a - (b*c)/d]*CosIntegral[(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)/b
3.1.63.3.1 Defintions of rubi rules used
Int[((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(p_.)*Sin[(c_.) + (d _.)*(x_)]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[(e + f*x)^m, Sin[a + b*x ]^p*Sin[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ[q, 0] && IntegerQ[m]
Int[((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*Cos[a + b*x]*(SinIntegral[c + d*x]/b), x] + (Simp[d/b Int[(e + f*x)^m*Cos[a + b*x]*(Sin[c + d*x]/(c + d*x)), x], x] + Simp[f*(m/b) Int[(e + f*x)^(m - 1)*Cos[a + b*x]*SinIntegr al[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
Int[Cos[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[Sin[a + b*x]*(SinIntegral[c + d*x]/b), x] - Simp[d/b Int[Sin[a + b*x] *(Sin[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(1245\) vs. \(2(351)=702\).
Time = 2.88 (sec) , antiderivative size = 1246, normalized size of antiderivative = 3.36
(-Si(d*x+c)/b*(-d/b*a*cos(1/d*b*(d*x+c)+(a*d-b*c)/d)-1/b*d*(sin(1/d*b*(d*x +c)+(a*d-b*c)/d)-(1/d*b*(d*x+c)+(a*d-b*c)/d)*cos(1/d*b*(d*x+c)+(a*d-b*c)/d )))+1/b*(1/2*a*d^2/(b-d)*(-Si(-(b-d)/d*(d*x+c)-(a*d-b*c)/d-(-a*d+b*c)/d)*c os((-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*c/(b-d)*(-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*(a*d-b*c)*d/(b-d)*(-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/(b-d)*d*cos((b-d)/d*(d*x+c)+(a*d-b*c)/d)-1/2*a*d^ 2/(b+d)*(-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*c/(b+d)*(-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* (a*d-b*c)*d/(b+d)*(-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/(b+d)*d*cos((b+d)/d*(d*x+c)+(a*d-b*c)/d)-1/2/b*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^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)))/d
Time = 0.29 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.16 \[ \int x \sin (a+b x) \text {Si}(c+d x) \, dx=\frac {2 \, b d^{2} \cos \left (b x + a\right ) \cos \left (d x + c\right ) + 2 \, b^{2} d \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 2 \, {\left (b^{3} d - b d^{3}\right )} x \cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) + 2 \, {\left (b^{2} d - d^{3}\right )} \sin \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) + {\left ({\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Si}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - {\left ({\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) + {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (b^{4} d - b^{2} d^{3}\right )}} \]
1/2*(2*b*d^2*cos(b*x + a)*cos(d*x + c) + 2*b^2*d*sin(b*x + a)*sin(d*x + c) - 2*(b^3*d - b*d^3)*x*cos(b*x + a)*sin_integral(d*x + c) + 2*(b^2*d - d^3 )*sin(b*x + a)*sin_integral(d*x + c) + ((b^2*d - d^3)*cos_integral((b*c + c*d + (b*d + d^2)*x)/d) - (b^2*d - d^3)*cos_integral(-(b*c - c*d + (b*d - d^2)*x)/d) - (b^3*c - b*c*d^2)*sin_integral((b*c + c*d + (b*d + d^2)*x)/d) - (b^3*c - b*c*d^2)*sin_integral(-(b*c - c*d + (b*d - d^2)*x)/d))*cos(-(b *c - a*d)/d) - ((b^3*c - b*c*d^2)*cos_integral((b*c + c*d + (b*d + d^2)*x) /d) - (b^3*c - b*c*d^2)*cos_integral(-(b*c - c*d + (b*d - d^2)*x)/d) + (b^ 2*d - d^3)*sin_integral((b*c + c*d + (b*d + d^2)*x)/d) + (b^2*d - d^3)*sin _integral(-(b*c - c*d + (b*d - d^2)*x)/d))*sin(-(b*c - a*d)/d))/(b^4*d - b ^2*d^3)
\[ \int x \sin (a+b x) \text {Si}(c+d x) \, dx=\int x \sin {\left (a + b x \right )} \operatorname {Si}{\left (c + d x \right )}\, dx \]
\[ \int x \sin (a+b x) \text {Si}(c+d x) \, dx=\int { x \sin \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) \,d x } \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.93 (sec) , antiderivative size = 200182, normalized size of antiderivative = 539.57 \[ \int x \sin (a+b x) \text {Si}(c+d x) \, dx=\text {Too large to display} \]
-(x*cos(b*x + a)/b - sin(b*x + a)/b^2)*sin_integral(d*x + c) - 1/4*(b^3*c* imag_part(cos_integral(b*x + d*x + c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*ta n(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*( b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - b*c*d^2*imag_part(cos_integral( b*x + d*x + c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2* tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2 *(b*c - c*d)/d)^2 - b^3*c*imag_part(cos_integral(b*x - d*x - c + b*c/d))*t an(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan( 1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + b*c*d ^2*imag_part(cos_integral(b*x - d*x - c + b*c/d))*tan(1/2*b*x + 1/2*d*x)^2 *tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/ 2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + b^3*c*imag_part(cos_integral (-b*x + d*x + c - b*c/d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^ 2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1 /2*(b*c - c*d)/d)^2 - b*c*d^2*imag_part(cos_integral(-b*x + d*x + c - b*c/ d))*tan(1/2*b*x + 1/2*d*x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2 *tan(1/2*a - 1/2*c)^2*tan(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 - b^3*c*imag_part(cos_integral(-b*x - d*x - c - b*c/d))*tan(1/2*b*x + 1/2*d* x)^2*tan(1/2*b*x - 1/2*d*x)^2*tan(1/2*a + 1/2*c)^2*tan(1/2*a - 1/2*c)^2*ta n(1/2*(b*c + c*d)/d)^2*tan(1/2*(b*c - c*d)/d)^2 + b*c*d^2*imag_part(cos...
Timed out. \[ \int x \sin (a+b x) \text {Si}(c+d x) \, dx=\int x\,\mathrm {sinint}\left (c+d\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]