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

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

[Out]

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

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Rubi [A]  time = 1.02, antiderivative size = 371, normalized size of antiderivative = 1.00, 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 2637

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

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 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 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]

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 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 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 6742

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

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*}

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Mathematica [C]  time = 7.06, size = 328, normalized size = 0.88 \[ \frac {-4 \text {Ci}(c+d x) (b x \cos (a+b x)-\sin (a+b x))+\frac {e^{-i a} \left (-(b c+i d) e^{\frac {i b c}{d}} \text {Ei}\left (-\frac {i (b-d) (c+d x)}{d}\right )-(b c+i d) e^{\frac {i b c}{d}} \text {Ei}\left (-\frac {i (b+d) (c+d x)}{d}\right )+i b e^{-i c} d \left (\frac {e^{i (-b x+2 c+d x)}}{b-d}+\frac {e^{-i x (b+d)}}{b+d}\right )\right )}{d}-\frac {e^{i (a-c)} \left ((b c-i d) e^{-\frac {i c (b-d)}{d}} \text {Ei}\left (\frac {i (b-d) (c+d x)}{d}\right )+(b c-i d) e^{-\frac {i c (b-d)}{d}} \text {Ei}\left (\frac {i (b+d) (c+d x)}{d}\right )+i b d \left (\frac {e^{i (x (b+d)+2 c)}}{b+d}+\frac {e^{i x (b-d)}}{b-d}\right )\right )}{d}}{4 b^2} \]

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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fricas [F]  time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Ci}\left (d x + c\right ) \sin \left (b x + a\right ), x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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maple [B]  time = 0.07, size = 1208, normalized size = 3.26 \[ \text {result too large to display} \]

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*(sin(b/d*(d*x+c)+(a*d-b*c)/d)-(b/d*(d*x+c)+(a*d-b*c)/d)*cos(b/d*(d*x+c)+(a*d-b*c)/d))+d/b*
a*cos(b/d*(d*x+c)+(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)*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)*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)*(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)*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)*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/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 = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Ci}\left (d x + c\right ) \sin \left (b x + a\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\mathrm {cosint}\left (c+d\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosint(c + d*x)*sin(a + b*x),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sin {\left (a + b x \right )} \operatorname {Ci}{\left (c + d x \right )}\, dx \]

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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