Integrand size = 14, antiderivative size = 370 \[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}+\frac {\operatorname {CosIntegral}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\operatorname {CosIntegral}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\sin (a-c+(b-d) x)}{2 b (b-d)}+\frac {\sin (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d} \] Output:
1/2*c*cos(a-b*c/d)*Ci(c*(b-d)/d+(b-d)*x)/b/d-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^2-1/2*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+cos(b*x+a)*Si(d*x+c)/b^2+x*sin(b*x+a)*Si(d*x+c)/ b-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
Result contains complex when optimal does not.
Time = 3.24 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.05 \[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=-\frac {e^{-i a} \left (-\left ((b c-i d) e^{2 i a-\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {i (b-d) (c+d x)}{d}\right )\right )+\frac {e^{-\frac {i (b+d) (c+d x)}{d}} \left (-i b d e^{\frac {i b c}{d}} \left (d \left (-1+e^{2 i (a+b x)}\right )+b \left (1+e^{2 i (a+b x)}\right )\right )+(b c+i d) \left (b^2-d^2\right ) e^{i \left (c+\frac {2 b c}{d}+(b+d) x\right )} \operatorname {ExpIntegralEi}\left (-\frac {i (b+d) (c+d x)}{d}\right )\right )}{(b-d) (b+d)}\right )}{4 b^2 d}+\frac {e^{-i a} \left (-\frac {i b d e^{i (c+(-b+d) x)} \left (b+d+b e^{2 i (a+b x)}-d e^{2 i (a+b x)}\right )}{(b-d) (b+d)}+(b c+i d) e^{\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {i (b-d) (c+d x)}{d}\right )-(b c-i d) e^{2 i a-\frac {i b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {i (b+d) (c+d x)}{d}\right )\right )}{4 b^2 d}+\frac {(\cos (a+b x)+b x \sin (a+b x)) \text {Si}(c+d x)}{b^2} \] Input:
Integrate[x*Cos[a + b*x]*SinIntegral[c + d*x],x]
Output:
-1/4*(-((b*c - I*d)*E^((2*I)*a - (I*b*c)/d)*ExpIntegralEi[(I*(b - d)*(c + d*x))/d]) + ((-I)*b*d*E^((I*b*c)/d)*(d*(-1 + E^((2*I)*(a + b*x))) + b*(1 + E^((2*I)*(a + b*x)))) + (b*c + I*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)))/(b^2*d*E^(I*a)) + (((-I)*b*d*E^(I*(c + (-b + d)*x ))*(b + d + b*E^((2*I)*(a + b*x)) - d*E^((2*I)*(a + b*x))))/((b - d)*(b + d)) + (b*c + I*d)*E^((I*b*c)/d)*ExpIntegralEi[((-I)*(b - d)*(c + d*x))/d] - (b*c - I*d)*E^((2*I)*a - (I*b*c)/d)*ExpIntegralEi[(I*(b + d)*(c + d*x))/ d])/(4*b^2*d*E^(I*a)) + ((Cos[a + b*x] + b*x*Sin[a + b*x])*SinIntegral[c + d*x])/b^2
Time = 1.75 (sec) , antiderivative size = 378, 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 = {7073, 5119, 2009, 7065, 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.
| \(\displaystyle \int x \cos (a+b x) \text {Si}(c+d x) \, dx\) |
| \(\Big \downarrow \) 7073 |
| \(\displaystyle -\frac {\int \sin (a+b x) \text {Si}(c+d x)dx}{b}-\frac {d \int \frac {x \sin (a+b x) \sin (c+d x)}{c+d x}dx}{b}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 5119 |
| \(\displaystyle -\frac {\int \sin (a+b x) \text {Si}(c+d x)dx}{b}-\frac {d \int \left (\frac {x \cos (a-c+(b-d) x)}{2 (c+d x)}-\frac {x \cos (a+c+(b+d) x)}{2 (c+d x)}\right )dx}{b}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\int \sin (a+b x) \text {Si}(c+d x)dx}{b}-\frac {d \left (-\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {\sin (a+x (b-d)-c)}{2 d (b-d)}-\frac {\sin (a+x (b+d)+c)}{2 d (b+d)}\right )}{b}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 7065 |
| \(\displaystyle -\frac {\frac {d \int \frac {\cos (a+b x) \sin (c+d x)}{c+d x}dx}{b}-\frac {\cos (a+b x) \text {Si}(c+d x)}{b}}{b}-\frac {d \left (-\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {\sin (a+x (b-d)-c)}{2 d (b-d)}-\frac {\sin (a+x (b+d)+c)}{2 d (b+d)}\right )}{b}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 4930 |
| \(\displaystyle -\frac {\frac {d \int \left (\frac {\sin (a+c+(b+d) x)}{2 (c+d x)}-\frac {\sin (a-c+(b-d) x)}{2 (c+d x)}\right )dx}{b}-\frac {\cos (a+b x) \text {Si}(c+d x)}{b}}{b}-\frac {d \left (-\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {\sin (a+x (b-d)-c)}{2 d (b-d)}-\frac {\sin (a+x (b+d)+c)}{2 d (b+d)}\right )}{b}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d \left (-\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d^2}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d^2}+\frac {\sin (a+x (b-d)-c)}{2 d (b-d)}-\frac {\sin (a+x (b+d)+c)}{2 d (b+d)}\right )}{b}-\frac {\frac {d \left (-\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 d}\right )}{b}-\frac {\cos (a+b x) \text {Si}(c+d x)}{b}}{b}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}\) |
Input:
Int[x*Cos[a + b*x]*SinIntegral[c + d*x],x]
Output:
(x*Sin[a + b*x]*SinIntegral[c + d*x])/b - (d*(-1/2*(c*Cos[a - (b*c)/d]*Cos Integral[(c*(b - d))/d + (b - d)*x])/d^2 + (c*Cos[a - (b*c)/d]*CosIntegral [(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]*SinIntegral [(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 - (-((Cos[a + b*x]*SinIntegral[c + d* x])/b) + (d*(-1/2*(CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d] )/d + (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*d) - (C os[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
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]
Int[(u_.)*Sin[(a_.) + (b_.)*(x_)]^(m_.)*Sin[(c_.) + (d_.)*(x_)]^(n_.), x_Sy mbol] :> Int[ExpandTrigReduce[u, Sin[a + b*x]^m*Sin[c + d*x]^n, x], x] /; F reeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Int[Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[(-Cos[a + b*x])*(SinIntegral[c + d*x]/b), x] + Simp[d/b Int[Cos[a + b *x]*(Sin[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
Int[Cos[(a_.) + (b_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e + f*x)^m*Sin[a + b*x]*(SinIntegral[c + d* x]/b), x] + (-Simp[d/b Int[(e + f*x)^m*Sin[a + b*x]*(Sin[c + d*x]/(c + d* x)), x], x] - Simp[f*(m/b) Int[(e + f*x)^(m - 1)*Sin[a + b*x]*SinIntegral [c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1207\) vs. \(2(350)=700\).
Time = 1.44 (sec) , antiderivative size = 1208, normalized size of antiderivative = 3.26
\[\text {Expression too large to display}\]
Input:
int(x*cos(b*x+a)*Si(d*x+c),x)
Output:
(Si(d*x+c)/b*(1/b*d*(cos(b*(d*x+c)/d+(a*d-b*c)/d)+(b*(d*x+c)/d+(a*d-b*c)/d )*sin(b*(d*x+c)/d+(a*d-b*c)/d))-d/b*a*sin(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)*(S i((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
Time = 0.11 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.16 \[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=\frac {2 \, b^{2} d \cos \left (b x + a\right ) \sin \left (d x + c\right ) + 2 \, {\left (b^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) - {\left ({\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) + {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) + {\left (b^{2} d - d^{3}\right )} \operatorname {Si}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - 2 \, {\left (b d^{2} \cos \left (d x + c\right ) - {\left (b^{3} d - b d^{3}\right )} x \operatorname {Si}\left (d x + c\right )\right )} \sin \left (b x + a\right ) - {\left ({\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{2} d - d^{3}\right )} \operatorname {Ci}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Si}\left (\frac {b c + c d + {\left (b d + d^{2}\right )} x}{d}\right ) - {\left (b^{3} c - b c d^{2}\right )} \operatorname {Si}\left (-\frac {b c - c d + {\left (b d - d^{2}\right )} x}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (b^{4} d - b^{2} d^{3}\right )}} \] Input:
integrate(x*cos(b*x+a)*sin_integral(d*x+c),x, algorithm="fricas")
Output:
1/2*(2*b^2*d*cos(b*x + a)*sin(d*x + c) + 2*(b^2*d - d^3)*cos(b*x + a)*sin_ integral(d*x + c) - ((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))*cos(-(b*c - a*d)/d) - 2*(b *d^2*cos(d*x + c) - (b^3*d - b*d^3)*x*sin_integral(d*x + c))*sin(b*x + a) - ((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_in tegral((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))*sin(-(b*c - a*d)/d))/(b^4*d - b^2*d^3)
\[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=\int x \cos {\left (a + b x \right )} \operatorname {Si}{\left (c + d x \right )}\, dx \] Input:
integrate(x*cos(b*x+a)*Si(d*x+c),x)
Output:
Integral(x*cos(a + b*x)*Si(c + d*x), x)
\[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=\int { x \cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ) \,d x } \] Input:
integrate(x*cos(b*x+a)*sin_integral(d*x+c),x, algorithm="maxima")
Output:
integrate(x*cos(b*x + a)*sin_integral(d*x + c), x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.58 (sec) , antiderivative size = 206132, normalized size of antiderivative = 557.11 \[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=\text {Too large to display} \] Input:
integrate(x*cos(b*x+a)*sin_integral(d*x+c),x, algorithm="giac")
Output:
(x*sin(b*x + a)/b + cos(b*x + a)/b^2)*sin_integral(d*x + c) - 1/4*(b^3*c*r eal_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*real_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*t an(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*real_part(cos_integral(b*x - d*x - c + b*c/d))*ta n(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*real_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*real_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*real_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*real_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*real_part(cos_...
Timed out. \[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=\int x\,\mathrm {sinint}\left (c+d\,x\right )\,\cos \left (a+b\,x\right ) \,d x \] Input:
int(x*sinint(c + d*x)*cos(a + b*x),x)
Output:
int(x*sinint(c + d*x)*cos(a + b*x), x)
\[ \int x \cos (a+b x) \text {Si}(c+d x) \, dx=\int \cos \left (b x +a \right ) \mathit {si} \left (d x +c \right ) x d x \] Input:
int(x*cos(b*x+a)*Si(d*x+c),x)
Output:
int(cos(a + b*x)*si(c + d*x)*x,x)