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

Optimal. Leaf size=371 \[ -\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b^2}-\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b^2}+\frac{\sin (a+b x) \text{CosIntegral}(c+d x)}{b^2}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b d}-\frac{x \cos (a+b x) \text{CosIntegral}(c+d x)}{b}-\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b d}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}+\frac{\sin (a+x (b-d)-c)}{2 b (b-d)}+\frac{\sin (a+x (b+d)+c)}{2 b (b+d)} \]

[Out]

-(c*Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (x*Cos[a + b*x]*CosIntegral[c + d*x])/b
- (c*Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) - (CosIntegral[(c*(b - d))/d + (b - d)*
x]*Sin[a - (b*c)/d])/(2*b^2) - (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b^2) + (CosIntegra
l[c + d*x]*Sin[a + b*x])/b^2 + Sin[a - c + (b - d)*x]/(2*b*(b - d)) + Sin[a + c + (b + d)*x]/(2*b*(b + d)) - (
Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (c*Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))
/d + (b - d)*x])/(2*b*d) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*Sin[a - (b*c
)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)

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Rubi [A]  time = 1.01904, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6520, 4609, 6742, 2637, 3303, 3299, 3302, 6512, 4430} \[ -\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b^2}-\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b^2}+\frac{\sin (a+b x) \text{CosIntegral}(c+d x)}{b^2}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b^2}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b^2}-\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b-d)}{d}+x (b-d)\right )}{2 b d}-\frac{x \cos (a+b x) \text{CosIntegral}(c+d x)}{b}-\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{c (b+d)}{d}+x (b+d)\right )}{2 b d}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b-d)+\frac{c (b-d)}{d}\right )}{2 b d}+\frac{c \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (x (b+d)+\frac{c (b+d)}{d}\right )}{2 b d}+\frac{\sin (a+x (b-d)-c)}{2 b (b-d)}+\frac{\sin (a+x (b+d)+c)}{2 b (b+d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (x*Cos[a + b*x]*CosIntegral[c + d*x])/b
- (c*Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) - (CosIntegral[(c*(b - d))/d + (b - d)*
x]*Sin[a - (b*c)/d])/(2*b^2) - (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b^2) + (CosIntegra
l[c + d*x]*Sin[a + b*x])/b^2 + Sin[a - c + (b - d)*x]/(2*b*(b - d)) + Sin[a + c + (b + d)*x]/(2*b*(b + d)) - (
Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (c*Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))
/d + (b - d)*x])/(2*b*d) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*Sin[a - (b*c
)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)

Rule 6520

Int[CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[((e
+ f*x)^m*Cos[a + b*x]*CosIntegral[c + d*x])/b, x] + (Dist[d/b, Int[((e + f*x)^m*Cos[a + b*x]*Cos[c + d*x])/(c
+ d*x), x], x] + Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Cos[a + b*x]*CosIntegral[c + d*x], x], x]) /; FreeQ[{a,
b, c, d, e, f}, x] && IGtQ[m, 0]

Rule 4609

Int[Cos[(a_.) + (b_.)*(x_)]^(m_.)*Cos[(c_.) + (d_.)*(x_)]^(n_.)*(u_.), x_Symbol] :> Int[ExpandTrigReduce[u, Co
s[a + b*x]^m*Cos[c + d*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 6512

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(Sin[a + b*x]*CosIntegral[c + d
*x])/b, x] - Dist[d/b, Int[(Sin[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 4430

Int[Cos[(c_.) + (d_.)*(x_)]^(q_.)*((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]

Rubi steps

\begin{align*} \int x \text{Ci}(c+d x) \sin (a+b x) \, dx &=-\frac{x \cos (a+b x) \text{Ci}(c+d x)}{b}+\frac{\int \cos (a+b x) \text{Ci}(c+d x) \, dx}{b}+\frac{d \int \frac{x \cos (a+b x) \cos (c+d x)}{c+d x} \, dx}{b}\\ &=-\frac{x \cos (a+b x) \text{Ci}(c+d x)}{b}+\frac{\text{Ci}(c+d x) \sin (a+b x)}{b^2}-\frac{d \int \frac{\cos (c+d x) \sin (a+b x)}{c+d x} \, dx}{b^2}+\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 \cos (a+b x) \text{Ci}(c+d x)}{b}+\frac{\text{Ci}(c+d x) \sin (a+b x)}{b^2}-\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^2}+\frac{d \int \frac{x \cos (a-c+(b-d) x)}{c+d x} \, dx}{2 b}+\frac{d \int \frac{x \cos (a+c+(b+d) x)}{c+d x} \, dx}{2 b}\\ &=-\frac{x \cos (a+b x) \text{Ci}(c+d x)}{b}+\frac{\text{Ci}(c+d x) \sin (a+b x)}{b^2}-\frac{d \int \frac{\sin (a-c+(b-d) x)}{c+d x} \, dx}{2 b^2}-\frac{d \int \frac{\sin (a+c+(b+d) x)}{c+d x} \, dx}{2 b^2}+\frac{d \int \left (\frac{\cos (a-c+(b-d) x)}{d}-\frac{c \cos (a-c+(b-d) x)}{d (c+d x)}\right ) \, dx}{2 b}+\frac{d \int \left (\frac{\cos (a+c+(b+d) x)}{d}-\frac{c \cos (a+c+(b+d) x)}{d (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac{x \cos (a+b x) \text{Ci}(c+d x)}{b}+\frac{\text{Ci}(c+d x) \sin (a+b x)}{b^2}+\frac{\int \cos (a-c+(b-d) x) \, dx}{2 b}+\frac{\int \cos (a+c+(b+d) x) \, dx}{2 b}-\frac{c \int \frac{\cos (a-c+(b-d) x)}{c+d x} \, dx}{2 b}-\frac{c \int \frac{\cos (a+c+(b+d) x)}{c+d x} \, dx}{2 b}-\frac{\left (d \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b^2}-\frac{\left (d \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2}-\frac{\left (d \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b^2}-\frac{\left (d \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2}\\ &=-\frac{x \cos (a+b x) \text{Ci}(c+d x)}{b}-\frac{\text{Ci}\left (\frac{c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac{b c}{d}\right )}{2 b^2}-\frac{\text{Ci}\left (\frac{c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac{b c}{d}\right )}{2 b^2}+\frac{\text{Ci}(c+d x) \sin (a+b x)}{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{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{c (b+d)}{d}+(b+d) x\right )}{2 b^2}-\frac{\left (c \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}-\frac{\left (c \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}+\frac{\left (c \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}+\frac{\left (c \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}\\ &=-\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{Ci}\left (\frac{c (b-d)}{d}+(b-d) x\right )}{2 b d}-\frac{x \cos (a+b x) \text{Ci}(c+d x)}{b}-\frac{c \cos \left (a-\frac{b c}{d}\right ) \text{Ci}\left (\frac{c (b+d)}{d}+(b+d) x\right )}{2 b d}-\frac{\text{Ci}\left (\frac{c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac{b c}{d}\right )}{2 b^2}-\frac{\text{Ci}\left (\frac{c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac{b c}{d}\right )}{2 b^2}+\frac{\text{Ci}(c+d x) \sin (a+b x)}{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 \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}\\ \end{align*}

Mathematica [C]  time = 6.40292, size = 328, normalized size = 0.88 \[ \frac{-4 \text{CosIntegral}(c+d x) (b x \cos (a+b x)-\sin (a+b x))+\frac{e^{-i a} \left (-(b c+i d) e^{\frac{i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b-d) (c+d x)}{d}\right )-(b c+i d) e^{\frac{i b c}{d}} \text{ExpIntegralEi}\left (-\frac{i (b+d) (c+d x)}{d}\right )+i b e^{-i c} d \left (\frac{e^{i (-b x+2 c+d x)}}{b-d}+\frac{e^{-i x (b+d)}}{b+d}\right )\right )}{d}-\frac{e^{i (a-c)} \left ((b c-i d) e^{-\frac{i c (b-d)}{d}} \text{ExpIntegralEi}\left (\frac{i (b-d) (c+d x)}{d}\right )+(b c-i d) e^{-\frac{i c (b-d)}{d}} \text{ExpIntegralEi}\left (\frac{i (b+d) (c+d x)}{d}\right )+i b d \left (\frac{e^{i (x (b+d)+2 c)}}{b+d}+\frac{e^{i x (b-d)}}{b-d}\right )\right )}{d}}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((I*b*d*(1/((b + d)*E^(I*(b + d)*x)) + E^(I*(2*c - b*x + d*x))/(b - d)))/E^(I*c) - (b*c + I*d)*E^((I*b*c)/d)*
ExpIntegralEi[((-I)*(b - d)*(c + d*x))/d] - (b*c + I*d)*E^((I*b*c)/d)*ExpIntegralEi[((-I)*(b + d)*(c + d*x))/d
])/(d*E^(I*a)) - (E^(I*(a - c))*(I*b*d*(E^(I*(b - d)*x)/(b - d) + E^(I*(2*c + (b + d)*x))/(b + d)) + ((b*c - I
*d)*ExpIntegralEi[(I*(b - d)*(c + d*x))/d])/E^((I*c*(b - d))/d) + ((b*c - I*d)*ExpIntegralEi[(I*(b + d)*(c + d
*x))/d])/E^((I*c*(b - d))/d)))/d - 4*CosIntegral[c + d*x]*(b*x*Cos[a + b*x] - Sin[a + b*x]))/(4*b^2)

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Maple [B]  time = 0.102, size = 1208, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Ci}\left (d x + c\right ) \sin \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{Ci}\left (d x + c\right ) \sin \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*cos_integral(d*x + c)*sin(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sin{\left (a + b x \right )} \operatorname{Ci}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out